# Cauchy sequence

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

A sequence in a metric space is Cauchy if, for every positive epsilon, all pairs of sufficiently late terms are less than epsilon apart.

mko-cauchy-sequence · definition · v0.10.0



## Mathematical statement

A sequence (x_n) is Cauchy if for every epsilon>0 there exists N such that all m,n≥N satisfy d(x_m,x_n)<epsilon. The definition does not first assume that a limit exists or belongs to the space.





## Representative formula (exact TeX)

C(x)





## Assumptions


- A finitely listed tail supplies only a bounded audit, not a proof of the general Cauchy property.
- Complete certificates are limited to an explicitly ultimately constant model, reciprocal-sequence bounds, or one exact counterexample family.





## Symbol scope


- C — Cauchy record
- x — sequence





## Intuition, definition, and misconceptions

The Cauchy condition compares every pair of late terms, not just each term with a proposed limit.

An ultimately constant sequence supplies a direct N for any positive epsilon.

Alternating positive and negative constants supply adjacent counterexample terms for every N.

A long finite tail with nearby terms does not establish that the whole infinite sequence is Cauchy.





## Adopted derivation guide

[Cauchy sequence: a complete certificate for the reciprocal sequence, a space in which it has no limit, an alternating counterexample and an ultimately constant model](https://ocme.evemisslab.com/en/derivations/derivation-mko-cauchy-sequence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Metric space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) — The metric supplies distances between terms.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — The Cauchy obligation quantifies over sequence indices.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Preserve the order: for every epsilon there exists N, for all m,n.





## Verification boundary

Finite tails are not general proofs; only explicit moduli or counterexample families are accepted.

not_formalized

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





## Next obligation

Add complete metric spaces and Banach spaces.





## 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/cauchy-sequence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-cauchy-sequence/index.md)