Complete metric space

Mathematical statement

A metric space (X,d) is complete if every Cauchy sequence in X converges to some point of X. For a finite space with distinct points, the least positive distance forces every Cauchy tail eventually to be constant; the singleton case is immediate. Testing a few sequences does not establish general completeness.

Representative formula (exact TeX)

C(X)

Assumptions

• The finite certificate uses a complete finite metric table on a carrier with at least one point; the minimum-positive-distance exercise protocol uses at least two.
• Sequence exercises accept only an explicit ultimately constant certificate or an explicit non-Cauchy counterexample; a finite prefix is not a universal completeness proof.

Symbol scope

• X — metric space
• d — metric
• x — sequence within the space

Intuition, definition, and misconceptions

Completeness is a property of the space, not of one selected sequence.

In a finite metric space, minimum positive separation gives a universal certificate, with the one-point case trivial.

A Cauchy sequence whose limit leaves a subspace can refute its completeness; this obligation is not replaced by a finite simulation here.

A few tested convergent sequences, or convergence in an ambient space, do not establish that the subspace is complete.

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

• Cauchy sequence — Completeness quantifies over all Cauchy sequences.
• Convergence in a metric space — The conclusion requires convergence inside the same space.
• Metric space — The complete metric supplies the distance obligations.

Verification boundary

A complete finite-metric certificate covers that finite space only; sampled sequences do not establish general-space completeness.

not_formalized

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

Next obligation

Together with norm-induced metrics, add real 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.