# Convergence in a metric space

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 converges to x when, for every positive epsilon, eventually all n≥N lie in the epsilon-ball around x.

mko-metric-convergence · definition · v0.10.0



## Mathematical statement

A sequence x_n in a metric space converges to x if, for every epsilon>0, there exists N such that all n≥N satisfy d(x_n,x)<epsilon. Passing a sampled finite tail is insufficient for general convergence; an explicitly ultimately constant infinite sequence can instead provide a complete witness.





## Representative formula (exact TeX)

L(x)





## Assumptions


- Exercises use complete finite metric tables.
- Finite-sequence questions are explicitly bounded audits; the ultimately constant model explicitly declares that every term after the prefix equals the tail value.





## Symbol scope


- L — limit record
- x — candidate limit





## Intuition, definition, and misconceptions

Open balls use the strict inequality d<epsilon.

A finite tail checks only recorded indices, not every future term.

An ultimately constant sequence permits N equal to the prefix length for every positive epsilon; metric separation supplies uniqueness of its limit.

Do not infer convergence because a long but finite tail appears close to a candidate.





## Adopted derivation guide

[Convergence in a metric space: limit and uniqueness for an ultimately constant sequence on a finite metric table](https://ocme.evemisslab.com/en/derivations/derivation-mko-metric-convergence-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/) — Epsilon-balls and distances come from a metric.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — Convergence quantifies over a naturally indexed sequence.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Retain the for every epsilon / exists N / for every later n obligations.





## Verification boundary

Only the explicitly ultimately constant model supplies a complete infinite witness in this slice.

not_formalized

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





## Next obligation

Add Cauchy sequences, completeness and continuous maps.





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