Convergence in a metric space

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Metric space — Epsilon-balls and distances come from a metric.
• Sequence — Convergence quantifies over a naturally indexed sequence.
• Quantifiers and explicit domains — 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.