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).
G001 — 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).
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.
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.
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.
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.
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.