# Monotone bounded sequence convergence theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A nondecreasing real sequence bounded above converges to the supremum of its terms; the nonincreasing bounded-below version converges to the infimum.

mko-monotone-sequence-convergence-theorem · theorem · v0.10.0



## Mathematical statement

Keep whole-sequence monotonicity and the corresponding bound. If s=sup{a_n}, s−epsilon is not an upper bound, supplying an index N; monotonicity then controls every later term a_n. The supremum or infimum need not be a term. A finite prefix supplies no universal certificate, and an unbounded monotone sequence or a bounded nonmonotone one is outside the theorem's hypotheses.





## Representative formula (exact TeX)

a_n\to s





## Assumptions


- The prototype uses the full symbolic tail a_n=L±c/(n+shift), c>0 and shift≥1; the denominator shift is separate from the theorem bound s.
- Epsilon is a positive rational and the error witness is strict.





## Symbol scope


- a_n — monotone real sequence
- s — supremum or infimum limit bound





## Intuition, definition, and misconceptions

Pair nondecreasing with an upper bound, and nonincreasing with a lower bound.

The supremum or infimum need not be attained.

Finite prefixes or a missing hypothesis alone do not supply this theorem certificate.

A few monotone terms do not prove whole-sequence convergence, and a supremum is not always a maximum term.





## Adopted derivation guide

[A bounded monotone tail and the supremum argument](https://ocme.evemisslab.com/en/derivations/derivation-mko-monotone-sequence-convergence-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Supremum and infimum](https://ocme.evemisslab.com/en/mko/mko-supremum-and-infimum/) — The appropriate bound is the limit candidate.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Convergence has an epsilon-tail obligation.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Monotonicity and bounds use the order.





## Verification boundary

Exact exercises use declared whole symbolic tails; finite samples do not prove the general theorem. The theorem bound and prototype denominator shift are kept distinct.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add Bolzano-Weierstrass and further compactness results.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/monotone-sequence-convergence-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-monotone-sequence-convergence-theorem/index.md)