Monotone bounded sequence convergence theorem

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

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 — The appropriate bound is the limit candidate.
• Sequence limits and exact tail cutoffs — Convergence has an epsilon-tail obligation.
• Usual order on the real numbers — 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.