Whole Monotone Tails, Strict Witnesses and Separate Hypothesis Evidence
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 1/2, 1, 2, [0, 1/6, 1/4], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
2. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 2/3, 2, 2, [-1/3, 0, 1/6], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
3. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 3/4, 3, 2, [-3/4, -1/4, 0], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
4. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 4/5, 4, 2, [-6/5, -8/15, -1/5], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
5. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 5/6, 5, 2, [-5/3, -5/6, -5/12], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
6. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 6/7, 6, 2, [-15/7, -8/7, -9/14], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
7. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 7/8, 7, 2, [-21/8, -35/24, -7/8], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
8. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 8/9, 8, 2, [-28/9, -16/9, -10/9], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
9. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 9/10, 9, 2, [-18/5, -21/10, -27/20], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
10. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 10/11, 10, 2, [-45/11, -80/33, -35/22], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
11. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 11/12, 11, 2, [-55/12, -11/4, -11/6], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
12. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 12/13, 12, 2, [-66/13, -40/13, -27/13], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
13. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 13/14, 13, 2, [-39/7, -143/42, -65/28], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
14. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 14/15, 14, 2, [-91/15, -56/15, -77/30], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
15. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 15/16, 15, 2, [-105/16, -65/16, -45/16], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
16. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 16/17, 16, 2, [-120/17, -224/51, -52/17], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
17. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 17/18, 17, 2, [-68/9, -85/18, -119/36], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
18. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 18/19, 18, 2, [-153/19, -96/19, -135/38], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
19. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 19/20, 19, 2, [-171/20, -323/60, -19/5], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
20. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 20/21, 20, 2, [-190/21, -40/7, -85/21], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
21. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 21/22, 21, 2, [-105/11, -133/22, -189/44], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
22. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 22/23, 22, 2, [-231/23, -440/69, -209/46], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
23. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 23/24, 23, 2, [-253/24, -161/24, -115/24], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
24. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 24/25, 24, 2, [-276/25, -176/25, -126/25], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
25. A Whole Nondecreasing Bounded Tail Has Its Supremum as Limit
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).
View answer and explanation
(increasing, 25/26, 25, 2, [-150/13, -575/78, -275/52], Yes, nondecreasing_bounded_above_supremum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-increasing-limit-certificate · candidate
26. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 1/2, 1, 2, [1, 5/6, 3/4], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
27. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 2/3, 2, 2, [5/3, 4/3, 7/6], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
28. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 3/4, 3, 2, [9/4, 7/4, 3/2], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
29. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 4/5, 4, 2, [14/5, 32/15, 9/5], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
30. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 5/6, 5, 2, [10/3, 5/2, 25/12], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
31. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 6/7, 6, 2, [27/7, 20/7, 33/14], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
32. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 7/8, 7, 2, [35/8, 77/24, 21/8], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
33. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 8/9, 8, 2, [44/9, 32/9, 26/9], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
34. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 9/10, 9, 2, [27/5, 39/10, 63/20], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
35. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 10/11, 10, 2, [65/11, 140/33, 75/22], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
36. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 11/12, 11, 2, [77/12, 55/12, 11/3], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
37. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 12/13, 12, 2, [90/13, 64/13, 51/13], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
38. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 13/14, 13, 2, [52/7, 221/42, 117/28], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
39. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 14/15, 14, 2, [119/15, 28/5, 133/30], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
40. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 15/16, 15, 2, [135/16, 95/16, 75/16], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
41. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 16/17, 16, 2, [152/17, 320/51, 84/17], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
42. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 17/18, 17, 2, [85/9, 119/18, 187/36], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
43. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 18/19, 18, 2, [189/19, 132/19, 207/38], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
44. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 19/20, 19, 2, [209/20, 437/60, 57/10], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
45. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 20/21, 20, 2, [230/21, 160/21, 125/21], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
46. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 21/22, 21, 2, [126/11, 175/22, 273/44], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
47. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 22/23, 22, 2, [275/23, 572/69, 297/46], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
48. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 23/24, 23, 2, [299/24, 69/8, 161/24], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
49. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 24/25, 24, 2, [324/25, 224/25, 174/25], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
50. A Whole Nonincreasing Bounded Tail Has Its Infimum as Limit
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).
View answer and explanation
(decreasing, 25/26, 25, 2, [175/13, 725/78, 375/52], Yes, nonincreasing_bounded_below_infimum_limit)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-decreasing-limit-certificate · candidate
51. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 1/2, 1/10, 9, 9/22, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
52. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 2/3, 1/11, 21, 52/69, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
53. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 3/4, 1/12, 35, 99/148, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
54. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 4/5, 1/13, 51, 232/265, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
55. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 5/6, 1/14, 69, 325/426, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
56. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 6/7, 1/15, 89, 12/13, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
57. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 7/8, 1/16, 111, 735/904, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
58. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 8/9, 1/17, 135, 1168/1233, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
59. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 9/10, 1/18, 161, 1377/1630, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
60. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 10/11, 1/19, 189, 2020/2101, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
61. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 11/12, 1/20, 219, 2299/2652, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
62. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 12/13, 1/21, 251, 3192/3289, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
63. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 13/14, 1/22, 285, 507/574, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
64. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 14/15, 1/23, 321, 4732/4845, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
65. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 15/16, 1/24, 359, 5175/5776, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
66. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 16/17, 1/25, 399, 6688/6817, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
67. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 17/18, 1/26, 441, 7225/7974, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
68. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 18/19, 1/27, 485, 9108/9253, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
69. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 19/20, 1/28, 531, 9747/10660, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
70. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 20/21, 1/29, 579, 1720/1743, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
71. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 21/22, 1/30, 629, 12789/13882, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
72. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 22/23, 1/31, 681, 15532/15709, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
73. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 23/24, 1/32, 735, 16399/17688, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
74. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(decreasing, 24/25, 1/33, 791, 19632/19825, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
75. A Strict Epsilon Tail Needs the First Passing Index, Including Equality Boundaries
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).
View answer and explanation
(increasing, 25/26, 1/34, 849, 20625/22126, Yes, forall_n_ge_N_by_monotone_symbolic_tail)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-epsilon-witness · candidate
76. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(monotone_unbounded, [0, 1, 2], Yes, No, No, monotone_without_bound_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
77. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(bounded_nonmonotone, [1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
78. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 5, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
79. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(monotone_unbounded, [0, 1, 2, 3, 4, 5], Yes, No, No, monotone_without_bound_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
80. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
81. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 8, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
82. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, No, monotone_without_bound_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
83. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1], No, Yes, No, bounded_without_monotonicity_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
84. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 11, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
85. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(monotone_unbounded, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, No, monotone_without_bound_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
86. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(bounded_nonmonotone, [1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1], No, Yes, No, bounded_without_monotonicity_not_covered)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
87. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 14, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
88. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
89. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
90. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 17, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
91. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
92. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
93. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 20, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
94. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
95. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
96. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 23, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
97. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
98. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
99. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(finite_prefix_only, 26, No, No, No, finite_prefix_not_universal_monotone_bounded_certificate)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
100. Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation
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).
For finite_prefix_only, No means no universal premise certificate was supplied, not that an unknown continuation lacks monotonicity, boundedness or convergence. The other modes specify whole counterexample families.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Monotone bounded sequence convergence theorem · qf-mod39-monoseq-hypothesis-boundary · candidate
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