{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0001","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b8076ab03c30906b1f096a1a0a0acfde39b47f21767e3189c0c961f52a02b60","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082001,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":1,\"limit\":[1,2],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=1/2, positive c=1 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 1/2, 1, 2, [0, 1/6, 1/4], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 1/2, 1, 2, [0, 1/6, 1/4], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0002","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e020361036c7f6bcb35b081ae117c7cce8adcd0a34fe5a47a9b80a3625e1c40d","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082002,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":2,\"limit\":[2,3],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=2/3, positive c=2 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 2/3, 2, 2, [-1/3, 0, 1/6], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 2/3, 2, 2, [-1/3, 0, 1/6], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0003","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d83a0640c4b21b2466aac3ec6b246a5469e76e8ea0280adba56c16e9c6d35110","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082003,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":3,\"limit\":[3,4],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=3/4, positive c=3 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 3/4, 3, 2, [-3/4, -1/4, 0], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 3/4, 3, 2, [-3/4, -1/4, 0], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0004","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da08a80b720383d55e6871c2b9c9148ab5c140ff6ae5b7d4332ab7f5c600dba4","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082004,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":4,\"limit\":[4,5],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=4/5, positive c=4 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 4/5, 4, 2, [-6/5, -8/15, -1/5], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 4/5, 4, 2, [-6/5, -8/15, -1/5], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0005","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c70cef54a81ed2fbf11e89b658e9e763b124861d9ac05e2752b0f0c994890aa","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082005,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":5,\"limit\":[5,6],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=5/6, positive c=5 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 5/6, 5, 2, [-5/3, -5/6, -5/12], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 5/6, 5, 2, [-5/3, -5/6, -5/12], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0006","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d31041e62e4797af9956de3b4b56b52129495cc75389cfd942baa709cbd6682e","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082006,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":6,\"limit\":[6,7],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=6/7, positive c=6 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 6/7, 6, 2, [-15/7, -8/7, -9/14], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 6/7, 6, 2, [-15/7, -8/7, -9/14], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0007","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa48ccb679c0f37ee30744b426fcc62582028b079af806d98cf44c046140d1ba","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082007,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":7,\"limit\":[7,8],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=7/8, positive c=7 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 7/8, 7, 2, [-21/8, -35/24, -7/8], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 7/8, 7, 2, [-21/8, -35/24, -7/8], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0008","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d997b10ffb27e1f9f5059134aa42af312fe3999de7055008da8cfd35c5c73ad3","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082008,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":8,\"limit\":[8,9],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=8/9, positive c=8 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 8/9, 8, 2, [-28/9, -16/9, -10/9], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 8/9, 8, 2, [-28/9, -16/9, -10/9], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0009","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ca5e767aa6390cd38508498167f5b2f57d2a4c8e97d9e6a2d80091c4ec2ea1b","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082009,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":9,\"limit\":[9,10],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=9/10, positive c=9 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 9/10, 9, 2, [-18/5, -21/10, -27/20], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 9/10, 9, 2, [-18/5, -21/10, -27/20], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0010","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d985c47d025e2f7499601ddb5b2151bdeb8c5d1593a77017dd17ba839645c9a9","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082010,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":10,\"limit\":[10,11],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=10/11, positive c=10 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 10/11, 10, 2, [-45/11, -80/33, -35/22], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 10/11, 10, 2, [-45/11, -80/33, -35/22], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0011","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40747e3cc07e4029ec04ad07fb567e3b68159387cd8131bfd25a560b84b3af2","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082011,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":11,\"limit\":[11,12],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=11/12, positive c=11 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 11/12, 11, 2, [-55/12, -11/4, -11/6], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 11/12, 11, 2, [-55/12, -11/4, -11/6], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0012","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b56ed22a110d4c07374c11b0b9fa440a3ca43b998de7051ed4ffbf9647d90fc","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082012,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":12,\"limit\":[12,13],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=12/13, positive c=12 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 12/13, 12, 2, [-66/13, -40/13, -27/13], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 12/13, 12, 2, [-66/13, -40/13, -27/13], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0013","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a99600031ee61ef0b800a472c288f0f9a13af086568bbb5d6a9f137968a2bce","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082013,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":13,\"limit\":[13,14],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=13/14, positive c=13 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 13/14, 13, 2, [-39/7, -143/42, -65/28], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 13/14, 13, 2, [-39/7, -143/42, -65/28], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0014","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44f427c26119fbc27494fe412bc31e895717887d1891ed498b9a7ea8431aa380","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082014,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":14,\"limit\":[14,15],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=14/15, positive c=14 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 14/15, 14, 2, [-91/15, -56/15, -77/30], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 14/15, 14, 2, [-91/15, -56/15, -77/30], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0015","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"449f1df18e4576a3107396d1e32e1861082a3e985279a0ae06cba4f4f5650a89","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082015,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":15,\"limit\":[15,16],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=15/16, positive c=15 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 15/16, 15, 2, [-105/16, -65/16, -45/16], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 15/16, 15, 2, [-105/16, -65/16, -45/16], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0016","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6239d936ca0da4bcaba6da2c364afe8b3181db95443651329b607b88a501b46f","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082016,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":16,\"limit\":[16,17],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=16/17, positive c=16 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 16/17, 16, 2, [-120/17, -224/51, -52/17], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 16/17, 16, 2, [-120/17, -224/51, -52/17], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0017","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9df6f418fca60521eba6017d59f3456bd7e7413884fa6a7cff0106246e453f73","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082017,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":17,\"limit\":[17,18],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=17/18, positive c=17 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 17/18, 17, 2, [-68/9, -85/18, -119/36], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 17/18, 17, 2, [-68/9, -85/18, -119/36], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0018","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b11d77204f309cf94145a1c81b8a96a6bfc42a7a8b6c1b51e6639a5eb01f6ad1","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082018,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":18,\"limit\":[18,19],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=18/19, positive c=18 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 18/19, 18, 2, [-153/19, -96/19, -135/38], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 18/19, 18, 2, [-153/19, -96/19, -135/38], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0019","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f6651398e11c449d47f3da412015b98b89000873f2f16ff5b831181dbd3746a","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082019,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":19,\"limit\":[19,20],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=19/20, positive c=19 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 19/20, 19, 2, [-171/20, -323/60, -19/5], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 19/20, 19, 2, [-171/20, -323/60, -19/5], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0020","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f16174160e3cfe4b1832dd43adb00899ae4b29e578013067efa7af88ef2a16e2","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082020,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":20,\"limit\":[20,21],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=20/21, positive c=20 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 20/21, 20, 2, [-190/21, -40/7, -85/21], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 20/21, 20, 2, [-190/21, -40/7, -85/21], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0021","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"977dfc38cce09eba1494bde522ecd1707cbe2e241de035ad23c6b880b5ce9d04","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082021,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":21,\"limit\":[21,22],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=21/22, positive c=21 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 21/22, 21, 2, [-105/11, -133/22, -189/44], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 21/22, 21, 2, [-105/11, -133/22, -189/44], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0022","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"536465f74e387a2720a9514dcf7cbc234ada37c2cd9fb418ad76e88a82cf75f4","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082022,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":22,\"limit\":[22,23],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=22/23, positive c=22 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 22/23, 22, 2, [-231/23, -440/69, -209/46], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 22/23, 22, 2, [-231/23, -440/69, -209/46], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0023","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f8e3aad84be320f11b10c1f67bca74bef21bc093e38177ad5edbf448bf3de88","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082023,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":23,\"limit\":[23,24],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=23/24, positive c=23 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 23/24, 23, 2, [-253/24, -161/24, -115/24], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 23/24, 23, 2, [-253/24, -161/24, -115/24], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0024","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63a374ef93bb30ce6aa761c36557c5280fbaf46124e8047d35adcf8627de2552","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082024,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":24,\"limit\":[24,25],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=24/25, positive c=24 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 24/25, 24, 2, [-276/25, -176/25, -126/25], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 24/25, 24, 2, [-276/25, -176/25, -126/25], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0025","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"152afdb452659cc107d5e20f38ca054cd508d16ac418afa1a79aa7a7a8075314","family_id":"qf-mod39-monoseq-increasing-limit-certificate","seed":202611082025,"semantic_template":"qf-mod39-monoseq-increasing-limit-certificate/{\"coefficient\":25,\"limit\":[25,26],\"shift\":2}"},"fields":{"title":"A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=25/26, positive c=25 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 25/26, 25, 2, [-150/13, -575/78, -275/52], Yes, nondecreasing_bounded_above_supremum_limit)","explanation":"Give the first three terms only as illustrations. Positive c and shift make every term below L and increase toward L; the supremum need not be attained. The symbolic family, not three samples, supplies the all-n monotonicity/bound/limit certificate. The exact result, retaining the requested order, is (increasing, 25/26, 25, 2, [-150/13, -575/78, -275/52], Yes, nondecreasing_bounded_above_supremum_limit)."},"skills":["A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0026","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8464e031609cb22a4d982d311fed652307f47653250c1e7b47652eee798433cc","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082026,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":1,\"limit\":[1,2],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=1/2, positive c=1 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 1/2, 1, 2, [1, 5/6, 3/4], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 1/2, 1, 2, [1, 5/6, 3/4], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0027","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2c6aac8d5ec70614d42fc8be938184dec13bf6ab903754a0766e7a73d72e41f","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082027,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":2,\"limit\":[2,3],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=2/3, positive c=2 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 2/3, 2, 2, [5/3, 4/3, 7/6], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 2/3, 2, 2, [5/3, 4/3, 7/6], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0028","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c99b2609f47a2ae491165e2d5c73d52645856535a4b4e8e7ffdd2d78bd08997f","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082028,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":3,\"limit\":[3,4],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=3/4, positive c=3 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 3/4, 3, 2, [9/4, 7/4, 3/2], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 3/4, 3, 2, [9/4, 7/4, 3/2], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0029","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24a12dd095ace046747a7453d2d57d8f01b5cb2f92e0c9d1f992e7387a31903f","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082029,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":4,\"limit\":[4,5],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=4/5, positive c=4 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 4/5, 4, 2, [14/5, 32/15, 9/5], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 4/5, 4, 2, [14/5, 32/15, 9/5], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0030","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd67134f9466e546ee5d76dffc344990187b7e743e1108681832cc54d376fbcb","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082030,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":5,\"limit\":[5,6],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=5/6, positive c=5 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 5/6, 5, 2, [10/3, 5/2, 25/12], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 5/6, 5, 2, [10/3, 5/2, 25/12], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0031","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e08c85aabb43d4242d8b3d99cb3e479e418678d96744f060fac829b212e88155","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082031,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":6,\"limit\":[6,7],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=6/7, positive c=6 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 6/7, 6, 2, [27/7, 20/7, 33/14], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 6/7, 6, 2, [27/7, 20/7, 33/14], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0032","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4392d920eb24c81804768f10b0f746f141ed00b4a8652487438368a707370f","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082032,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":7,\"limit\":[7,8],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=7/8, positive c=7 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 7/8, 7, 2, [35/8, 77/24, 21/8], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 7/8, 7, 2, [35/8, 77/24, 21/8], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0033","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c16ba7f00fa2043b7a8b99c372cab8e276033ac3294dff0544230e3e3a0e8f2","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082033,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":8,\"limit\":[8,9],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=8/9, positive c=8 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 8/9, 8, 2, [44/9, 32/9, 26/9], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 8/9, 8, 2, [44/9, 32/9, 26/9], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0034","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20e4b2c58d74d7f07f2f3db3aded00a9cd587042dbe023ea0208824e89af185c","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082034,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":9,\"limit\":[9,10],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=9/10, positive c=9 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 9/10, 9, 2, [27/5, 39/10, 63/20], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 9/10, 9, 2, [27/5, 39/10, 63/20], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0035","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb3e606b10f7d89370284cdb061af9b7b9086080e5e15ae4471d57f388584cf4","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082035,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":10,\"limit\":[10,11],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=10/11, positive c=10 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 10/11, 10, 2, [65/11, 140/33, 75/22], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 10/11, 10, 2, [65/11, 140/33, 75/22], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0036","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6652fe7403b2915efa2d41e1ab163240acaed90d8c64d45ca8edaaba3d1896d","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082036,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":11,\"limit\":[11,12],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=11/12, positive c=11 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 11/12, 11, 2, [77/12, 55/12, 11/3], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 11/12, 11, 2, [77/12, 55/12, 11/3], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0037","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b06afbc7a25889c324f0167072cd6fb351aea18b4fbcc6ca423a7a416b1e726e","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082037,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":12,\"limit\":[12,13],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=12/13, positive c=12 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 12/13, 12, 2, [90/13, 64/13, 51/13], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 12/13, 12, 2, [90/13, 64/13, 51/13], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0038","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c6209a882a8ea8c44191c8879da41452715a142e7031fcaa88e5e2c2fc19fa3","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082038,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":13,\"limit\":[13,14],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=13/14, positive c=13 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 13/14, 13, 2, [52/7, 221/42, 117/28], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 13/14, 13, 2, [52/7, 221/42, 117/28], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0039","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dcdcec69e34a79109157e26b7148a758f73d5a206add2533a1a34f9697ba6a8","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082039,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":14,\"limit\":[14,15],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=14/15, positive c=14 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 14/15, 14, 2, [119/15, 28/5, 133/30], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 14/15, 14, 2, [119/15, 28/5, 133/30], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0040","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bf174631621bd6c69c6d3ee1d3d044ea7059ea28ecea8dfbab8d32d1a29ae78","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082040,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":15,\"limit\":[15,16],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=15/16, positive c=15 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 15/16, 15, 2, [135/16, 95/16, 75/16], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 15/16, 15, 2, [135/16, 95/16, 75/16], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0041","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"249812964b8c14831155682db00a29f5a0fd2fc2257384244fc552f7600a5c96","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082041,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":16,\"limit\":[16,17],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=16/17, positive c=16 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 16/17, 16, 2, [152/17, 320/51, 84/17], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 16/17, 16, 2, [152/17, 320/51, 84/17], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0042","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e02070711bb52ada1d770a670cb46d8817d138dbe6c75fbec62b976f68d3839","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082042,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":17,\"limit\":[17,18],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=17/18, positive c=17 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 17/18, 17, 2, [85/9, 119/18, 187/36], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 17/18, 17, 2, [85/9, 119/18, 187/36], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0043","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d7327a0e748f82f675a550c4b910a580689eef05a909d27b6a1be5047e7478e","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082043,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":18,\"limit\":[18,19],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=18/19, positive c=18 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 18/19, 18, 2, [189/19, 132/19, 207/38], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 18/19, 18, 2, [189/19, 132/19, 207/38], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0044","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10be95932aa06ecdfbb8aca84cee215392af19920e308418f1a9cee11a305147","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082044,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":19,\"limit\":[19,20],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=19/20, positive c=19 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 19/20, 19, 2, [209/20, 437/60, 57/10], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 19/20, 19, 2, [209/20, 437/60, 57/10], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0045","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5dc9f9530c407485e9b9744dddbb41e783b7a3a98a2946d72d08a1328ac1567","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082045,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":20,\"limit\":[20,21],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=20/21, positive c=20 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 20/21, 20, 2, [230/21, 160/21, 125/21], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 20/21, 20, 2, [230/21, 160/21, 125/21], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0046","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd7b12268e8ee41962bb54fa8e1bf8edc21ef9b0d7a1d035972a1535f1ef8411","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082046,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":21,\"limit\":[21,22],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=21/22, positive c=21 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 21/22, 21, 2, [126/11, 175/22, 273/44], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 21/22, 21, 2, [126/11, 175/22, 273/44], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0047","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ef62bdb61a082fa5cfa1a67a9fa9b29f6902183494ca9c2ea150d6f3bcd031","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082047,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":22,\"limit\":[22,23],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=22/23, positive c=22 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 22/23, 22, 2, [275/23, 572/69, 297/46], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 22/23, 22, 2, [275/23, 572/69, 297/46], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0048","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd63b3f0c18205be243ea4b4a93552e7e66f46b859874c28dbf825dfd327c98a","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082048,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":23,\"limit\":[23,24],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=23/24, positive c=23 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 23/24, 23, 2, [299/24, 69/8, 161/24], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 23/24, 23, 2, [299/24, 69/8, 161/24], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0049","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c5e665eaf8c7bfc4e75494607909c2af50b530e841f780cde9f7f1979b84403","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082049,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":24,\"limit\":[24,25],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=24/25, positive c=24 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 24/25, 24, 2, [324/25, 224/25, 174/25], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 24/25, 24, 2, [324/25, 224/25, 174/25], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0050","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58dee2e70e79a539d23be93639bc264b27f84056add95ad960878c7d6dd57587","family_id":"qf-mod39-monoseq-decreasing-limit-certificate","seed":202611082050,"semantic_template":"qf-mod39-monoseq-decreasing-limit-certificate/{\"coefficient\":25,\"limit\":[25,26],\"shift\":2}"},"fields":{"title":"A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=25/26, positive c=25 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Give (direction, limit, coefficient, shift, first_three_terms, symbolic_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 25/26, 25, 2, [175/13, 725/78, 375/52], Yes, nonincreasing_bounded_below_infimum_limit)","explanation":"Give the first three terms as illustrations. The whole declared family decreases from above L, is bounded below by L and converges to that infimum; the displayed samples alone are not the proof. The exact result, retaining the requested order, is (decreasing, 25/26, 25, 2, [175/13, 725/78, 375/52], Yes, nonincreasing_bounded_below_infimum_limit)."},"skills":["A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0051","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"690da4aa896f9689f9ba59407b1da6159f88ea8c53ab4e5177c12299768663da","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082051,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":1,\"direction\":\"increasing\",\"epsilon\":[1,10],\"limit\":[1,2],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=1/2, positive c=1 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/10. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 1/2, 1/10, 9, 9/22, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 1/2, 1/10, 9, 9/22, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0052","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59be09cbed84c700386bc94f4294f4836d9c3a814121700b079e73c91d03b89f","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082052,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":2,\"direction\":\"decreasing\",\"epsilon\":[1,11],\"limit\":[2,3],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=2/3, positive c=2 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/11. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 2/3, 1/11, 21, 52/69, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 2/3, 1/11, 21, 52/69, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0053","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fcf2a2cb025b546e237a86d458ebb2cf993d4be5d58d0f6b4b098c205f134d9","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082053,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":3,\"direction\":\"increasing\",\"epsilon\":[1,12],\"limit\":[3,4],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=3/4, positive c=3 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/12. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 3/4, 1/12, 35, 99/148, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 3/4, 1/12, 35, 99/148, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0054","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3702aada60b961c14abe0c74fc9ae63cb8ba3ba4b719ec9c807236773d0c0a8","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082054,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":4,\"direction\":\"decreasing\",\"epsilon\":[1,13],\"limit\":[4,5],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=4/5, positive c=4 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/13. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 4/5, 1/13, 51, 232/265, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 4/5, 1/13, 51, 232/265, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0055","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8c2659026297d054d8f655f5c49612d762ec3543af2e0ff46cfcb62b0cae0dc","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082055,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":5,\"direction\":\"increasing\",\"epsilon\":[1,14],\"limit\":[5,6],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=5/6, positive c=5 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/14. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 5/6, 1/14, 69, 325/426, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 5/6, 1/14, 69, 325/426, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0056","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbf120e45da089240e693d92ee59064e0ea5689c6962209379b4e26734bf9759","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082056,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":6,\"direction\":\"decreasing\",\"epsilon\":[1,15],\"limit\":[6,7],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=6/7, positive c=6 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/15. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 6/7, 1/15, 89, 12/13, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 6/7, 1/15, 89, 12/13, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0057","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c80ae1d04b63519fc48df8d78c95e4861344fb7091915052fc46d078e79ea341","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082057,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":7,\"direction\":\"increasing\",\"epsilon\":[1,16],\"limit\":[7,8],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=7/8, positive c=7 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/16. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 7/8, 1/16, 111, 735/904, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 7/8, 1/16, 111, 735/904, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0058","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2372b04f5d3e159b65c7e206545efecce4b82f4c150c86456944f1d2b5180d7c","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082058,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":8,\"direction\":\"decreasing\",\"epsilon\":[1,17],\"limit\":[8,9],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=8/9, positive c=8 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/17. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 8/9, 1/17, 135, 1168/1233, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 8/9, 1/17, 135, 1168/1233, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0059","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c110d21510f0a10fe8f9ea693be60fce3a86b18437d335863deada3d9e6fbb70","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082059,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":9,\"direction\":\"increasing\",\"epsilon\":[1,18],\"limit\":[9,10],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=9/10, positive c=9 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/18. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 9/10, 1/18, 161, 1377/1630, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 9/10, 1/18, 161, 1377/1630, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0060","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bc50a3764fd52f90da7c96775eb953aff62a0a189b8f482068778db60751463","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082060,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":10,\"direction\":\"decreasing\",\"epsilon\":[1,19],\"limit\":[10,11],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=10/11, positive c=10 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/19. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 10/11, 1/19, 189, 2020/2101, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 10/11, 1/19, 189, 2020/2101, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0061","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cc0a10772d84e9f6bad6d9b1aafd49beabc2b6f1c6cda3fa8e7455be18a528a","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082061,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":11,\"direction\":\"increasing\",\"epsilon\":[1,20],\"limit\":[11,12],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=11/12, positive c=11 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/20. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 11/12, 1/20, 219, 2299/2652, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 11/12, 1/20, 219, 2299/2652, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0062","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9c3be9c1ea7dbe426da42160b9bd122fc661f921300e4ff5bc859386479554e","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082062,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":12,\"direction\":\"decreasing\",\"epsilon\":[1,21],\"limit\":[12,13],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=12/13, positive c=12 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/21. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 12/13, 1/21, 251, 3192/3289, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 12/13, 1/21, 251, 3192/3289, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0063","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ffb9d42ed7c9a3325ac9356cd6ccf784bbadcdc9577fa1a993e624e37ef0ad1","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082063,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":13,\"direction\":\"increasing\",\"epsilon\":[1,22],\"limit\":[13,14],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=13/14, positive c=13 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/22. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 13/14, 1/22, 285, 507/574, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 13/14, 1/22, 285, 507/574, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0064","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"196a4e18e99399ca183e7c2497669a19a99e4ce5422c797603189281924afc1a","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082064,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":14,\"direction\":\"decreasing\",\"epsilon\":[1,23],\"limit\":[14,15],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=14/15, positive c=14 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/23. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 14/15, 1/23, 321, 4732/4845, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 14/15, 1/23, 321, 4732/4845, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0065","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"844673c94ab34310a94adef3cffedd24a40610b681f98263b7ccee2e4552c99c","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082065,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":15,\"direction\":\"increasing\",\"epsilon\":[1,24],\"limit\":[15,16],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=15/16, positive c=15 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/24. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 15/16, 1/24, 359, 5175/5776, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 15/16, 1/24, 359, 5175/5776, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0066","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f49e2b8657e2bcf6f4885844a38b07a7802ceb7455a79741ac21cde8cc7f0c3","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082066,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":16,\"direction\":\"decreasing\",\"epsilon\":[1,25],\"limit\":[16,17],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=16/17, positive c=16 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/25. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 16/17, 1/25, 399, 6688/6817, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 16/17, 1/25, 399, 6688/6817, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0067","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"278cdafdd3ae33b1a1af87f9ab2cbd8a7ca2fb1150f5555fff6e1f49ee861d47","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082067,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":17,\"direction\":\"increasing\",\"epsilon\":[1,26],\"limit\":[17,18],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=17/18, positive c=17 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/26. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 17/18, 1/26, 441, 7225/7974, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 17/18, 1/26, 441, 7225/7974, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0068","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e3029f406690d09d1907d018ef6bd8b299066aa3add9591c8ae9e0f6073d5a6","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082068,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":18,\"direction\":\"decreasing\",\"epsilon\":[1,27],\"limit\":[18,19],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=18/19, positive c=18 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/27. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 18/19, 1/27, 485, 9108/9253, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 18/19, 1/27, 485, 9108/9253, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0069","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59ec610eb48b4dee4adb8d23862acbc51490793c8849c28c8a7552e70488ca23","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082069,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":19,\"direction\":\"increasing\",\"epsilon\":[1,28],\"limit\":[19,20],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=19/20, positive c=19 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/28. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 19/20, 1/28, 531, 9747/10660, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 19/20, 1/28, 531, 9747/10660, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0070","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd95238d02cd71dedf7c493af7fabdf4edc98ecefab6063f30b396202d82ec57","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082070,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":20,\"direction\":\"decreasing\",\"epsilon\":[1,29],\"limit\":[20,21],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=20/21, positive c=20 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/29. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 20/21, 1/29, 579, 1720/1743, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 20/21, 1/29, 579, 1720/1743, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0071","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dba0ae9d9fa83c891e8ed3bce53dac4ff61f73e230b21553e70255b3d9b0a9e8","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082071,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":21,\"direction\":\"increasing\",\"epsilon\":[1,30],\"limit\":[21,22],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=21/22, positive c=21 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/30. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 21/22, 1/30, 629, 12789/13882, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 21/22, 1/30, 629, 12789/13882, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0072","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701aba62ee2b423b96703cf5c188bad07121952b59fd6ccfa7de69291e4dfa9f","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082072,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":22,\"direction\":\"decreasing\",\"epsilon\":[1,31],\"limit\":[22,23],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=22/23, positive c=22 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/31. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 22/23, 1/31, 681, 15532/15709, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 22/23, 1/31, 681, 15532/15709, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0073","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fd2eee9d310ba144101bd92bf095a7eab417228f931d25c81e4854e41e16a81","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082073,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":23,\"direction\":\"increasing\",\"epsilon\":[1,32],\"limit\":[23,24],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=23/24, positive c=23 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/32. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 23/24, 1/32, 735, 16399/17688, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 23/24, 1/32, 735, 16399/17688, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0074","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"148c5d2f0dc9b908e1b8de45aefe2ea7e5f6dac90437fdc3914380200b2b84bc","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082074,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":24,\"direction\":\"decreasing\",\"epsilon\":[1,33],\"limit\":[24,25],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L+c/(n+shift) for every n≥0, with L=24/25, positive c=24 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=decreasing, positive epsilon=1/33. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(decreasing, 24/25, 1/33, 791, 19632/19825, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (decreasing, 24/25, 1/33, 791, 19632/19825, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0075","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8d3eec269f1dc2082a20240798bc06c908ba6a85d5ee47110cc942a867b2ef1","family_id":"qf-mod39-monoseq-epsilon-witness","seed":202611082075,"semantic_template":"qf-mod39-monoseq-epsilon-witness/{\"coefficient\":25,\"direction\":\"increasing\",\"epsilon\":[1,34],\"limit\":[25,26],\"shift\":2}"},"fields":{"title":"A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries","stem":"Use the whole real sequence a_n=L−c/(n+shift) for every n≥0, with L=25/26, positive c=25 and positive integer shift=2. The denominator shift is distinct from the theorem's supremum/infimum bound. The entire symbolic family, not a displayed finite prefix, supplies monotonicity and the corresponding bound. Direction=increasing, positive epsilon=1/34. Give (direction, limit, epsilon, N, value_at_N, strict_tail_certificate, certificate).","answer":{"kind":"expression","value":"(increasing, 25/26, 1/34, 849, 20625/22126, Yes, forall_n_ge_N_by_monotone_symbolic_tail)","explanation":"Find the least nonnegative N with c/(N+shift)<epsilon. A gap equal to epsilon still fails the strict criterion. Report a_N and retain the all-n≥N tail flag; increasing denominators preserve the strict bound. The exact result, retaining the requested order, is (increasing, 25/26, 1/34, 849, 20625/22126, Yes, forall_n_ge_N_by_monotone_symbolic_tail)."},"skills":["A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0076","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c756ff17beb78c3aed7d75a4dd74ee1e762b91b585c4dc19dc57e1d65eca3ef","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082076,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":3}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=3. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0077","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"88225bda70962b8492bdaca1161638787ae4b33d8f5d801ad26b300a0113a566","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082077,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":4}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=4. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0078","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c6aa68be3da411c4cff566fe0c93348b83acff26efee31b0f800bc7aa9ba201","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082078,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":5}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=5. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 5, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 5, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0079","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63544bce9af45068cef31135ce6b7569c3629fbb7e0927314696581685e6bd8f","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082079,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":6}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=6. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0080","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13c11df6b509c4cfa1143d682d6a74bcefa4e5424c6adc65410f9524c8987fa4","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082080,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":7}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=7. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0081","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46560b7080489f9867bf5f19de89e0cca31e5a8e91893b9d50607084a78fff0d","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082081,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":8}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=8. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 8, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 8, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0082","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dc5b27c0534fbb5cb6328db096b206a40d2073a3884a83332cf3f3bbf8957a7","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082082,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":9}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=9. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0083","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed3dad9ef3c99c4d5d4d8c7e024629971eaf549311c0fdf1671b2429113c665c","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082083,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":10}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=10. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0084","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ff5e259ae6424549ee5adbf393a8105befc9fb9532da2185678c311991c09d7","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082084,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":11}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=11. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 11, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 11, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0085","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e287736cf1f9808c621fdcc3e0998fbbacfc88c5bd2bb3b14767b8946cff56a","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082085,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":12}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=12. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0086","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66b7d129b43a803e6adccf577436250ce35d3c3f04118917c75074265643f6eb","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082086,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":13}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=13. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0087","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30c64125ddb72e6cdb80d311cfcab96b093c1e12f464f74dc4b7dc4bbd649fd5","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082087,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":14}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=14. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 14, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 14, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0088","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a272ace9e708a89430d6048094fe2f4afff4e3a14e60366359e12d041449d79","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082088,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":15}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=15. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0089","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ad3dac83af6c93aa739193c4addd95051bd1547420d42dbf7917fc583691c94","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082089,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":16}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=16. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0090","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e7347fd4c44118b6f6796405fd39b2a8ea22b8dc4ea0957106e7a99a5371470","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082090,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":17}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=17. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 17, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 17, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0091","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"643ff21d16246fae475dbc8458b9261fc4d89340de57fa96580295994b4a623c","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082091,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":18}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=18. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0092","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8ab15aa6d0a86ab6584dcbf032fd85f3c8bd4d4120dad5b47501d7a7c51505b","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082092,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":19}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=19. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0093","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ad432c4cdd207b633b176b7181fbe0a2efd2d6d20f9d90f4925816878267724","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082093,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":20}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=20. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 20, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 20, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0094","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1fa649cd28bb3360288a5c5598c8f630d4cbe4e4ac4efb68afd5921ed8ff39e7","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082094,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":21}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=21. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0095","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3e3e445077620020f6af0a36e70eb07eb056323c4da86a94daa108ef69ffb4e","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082095,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":22}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=22. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0096","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf10f7a526ffe9d9edae2a1dfdfc0879b7b9d77bcfa89c03cbd113ea02a0a3f1","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082096,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":23}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=23. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 23, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 23, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0097","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9298a8534382214f114670d8918ef54d4f3ebfff65bac4dcb7751cfb8acc04ee","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082097,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":24}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=24. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0098","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c36ee91b4b7dbafc3fdc553b71e262f0242abe695586de705599a4f1d3528f4d","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082098,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"bounded_nonmonotone\",\"prefix_length\":25}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=bounded_nonmonotone, displayed prefix_length=25. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0099","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9bac0f1e0148d11d0254cadee9a21a6e1f127535fa52ef802907d201d6a6a8d","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082099,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"finite_prefix_only\",\"prefix_length\":26}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=finite_prefix_only, displayed prefix_length=26. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(finite_prefix_only, 26, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (finite_prefix_only, 26, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0100","batch_id":"batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c506d9acd1b0569afae7c453ec9164ec3f363b6bcdf47e8ebae70a0090acbd20","family_id":"qf-mod39-monoseq-hypothesis-boundary","seed":202611082100,"semantic_template":"qf-mod39-monoseq-hypothesis-boundary/{\"mode\":\"monotone_unbounded\",\"prefix_length\":27}"},"fields":{"title":"Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation","stem":"Boundary mode=monotone_unbounded, displayed prefix_length=27. For monotone_unbounded the declared full example is a_n=n; for bounded_nonmonotone it is a_n=(−1)^n, both on n≥0. finite_prefix_only supplies only a length, not a whole sequence. Give (mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary).","answer":{"kind":"expression","value":"(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], Yes, No, No, monotone_without_bound_not_covered)","explanation":"Keep monotonicity, the relevant bound and applicability of the monotone-bounded theorem separate. In finite-prefix mode the No values withhold universal certificates, not classify an unspecified continuation. Bounded nonmonotone sequences may have other convergence arguments; the displayed alternating example does not meet this theorem. The exact result, retaining the requested order, is (monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26], Yes, No, No, monotone_without_bound_not_covered)."},"skills":["Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}