# Bolzano-Weierstrass theorem for real sequences

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

Every whole bounded real sequence has a convergent subsequence selected by an infinite strictly increasing index map.

mko-bolzano-weierstrass-theorem-real · theorem · v0.10.0



## 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

[Two convergent subsequences of an oscillatory tail](https://ocme.evemisslab.com/en/derivations/derivation-mko-bolzano-weierstrass-theorem-real-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Subsequence](https://ocme.evemisslab.com/en/mko/mko-subsequence/) — The index map is strictly increasing.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Convergence has an epsilon-tail obligation.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — The bounded sequence is real-valued.
- [Supremum and infimum](https://ocme.evemisslab.com/en/mko/mko-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.

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