Bolzano-Weierstrass theorem for real sequences

Mathematical statement

Boundedness means there exists M bounding all terms, not just a finite prefix. A subsequence has an infinite strictly increasing index map, and convergence has an all-later epsilon-tail witness. Periodic sequences and symbolic oscillatory rational tails give exact instances here. A finite prefix or finite cluster search is insufficient. The theorem does not require the original sequence to converge or guarantee that every convergent subsequence has the same limit.

Representative formula (exact TeX)

a_k\to L

Assumptions

• The main families declare a complete periodic law or a whole symbolic oscillatory tail.
• Epsilon is positive rational, and the subsequence index map is affine and strictly increasing.

Symbol scope

• a_k — convergent subsequence
• L — subsequence limit

Intuition, definition, and misconceptions

Whole-sequence boundedness cannot be replaced by a finite prefix.

The subsequence index map must be infinite and strictly increasing.

The theorem guarantees some convergent subsequence, not convergence of the original sequence.

Repeated values in a finite observation do not prove the general Bolzano-Weierstrass theorem.

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

• Subsequence — The index map is strictly increasing.
• Sequence limits and exact tail cutoffs — Convergence has an epsilon-tail obligation.
• Real number — The bounded sequence is real-valued.
• Supremum and infimum — Completeness supports the existence argument.

Verification boundary

Exact question families certify only their declared symbolic sequences.

not_formalized

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

Next obligation

Add sequential compactness.

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.