# Question 21075: Missing Premise Certificates Are Not Negated Properties of an Unknown Continuation [Return to catalogue page 106](https://ocme.evemisslab.com/en/questions/page/106/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-foundations/#q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0100) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0041-21075 · q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0100 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-monotone-sequence-convergence-theorem](https://ocme.evemisslab.com/en/mko/mko-monotone-sequence-convergence-theorem/) ## Step-by-step derivation S001 · impliesBefore source operation=monoseq-hypothesis-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[mode, prefix_or_length, monotone_premise_certified, relevant_bound_certified, theorem_applicability_certified, boundary] Conditions: none Retain the historical interface marker while listing ordered output fields without JavaScript.Fields and tags are not extra mathematical evidence;fixed No markers retain their claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL mode=monotone_unbounded;displayed prefix length=27;prefix_or_length=[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] After declared WHOLE a_n=n:adjacent difference1>0;for ANY real bound B choose natural n>B,so not bounded above and not a finite-limit sequence Conditions: none Distinguish a whole-sequence formula from a finite length.In the first two cases,monotonicity,bounds and divergence follow from the declared infinite law and explicit witnesses,not the displayed prefix. Step identifiers and dependenciesRule: monoseq_boundary_whole_law_not_observed_prefix · Uses: G001, S001 S003 · impliesBefore ORIGINAL boundary mode=monotone_unbounded;actual certificate flags=(Yes, No, No) After monotone convergence theorem not certified here;convergence in GENERAL does NOT imply monotonicity;unknown continuation not declared divergent Conditions: none Monotonicity and the matching bound are sufficient for this theorem,not necessary for every convergent sequence.A finite-prefix No does not make all extensions divergent;the specified alternating example has its own even/odd divergence witness. Step identifiers and dependenciesRule: monoseq_hypotheses_sufficient_not_convergence_necessity · Uses: G001, S002 ## Reverse checks - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (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) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: c506d9acd1b0569afae7c453ec9164ec3f363b6bcdf47e8ebae70a0090acbd20 - candidate record SHA-256: 66817caf2fcd6cbd6ce97baf8e33b632b0f0b1c939cb3f817c25d1203c1348ec - semantic review receipt SHA-256: 8f134403487694af4c70d2e00ec84bab542043ef4b3c54517c04f3c409c0262a - question-first witness SHA-256: ad772845088ed8e99f87f90d4264b05ea97be35775305a02011af5ee35f0f4ce - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r02-monotone-sequence-convergence-theorem-0100.json)