Cauchy sequence

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

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

Prerequisites

• Metric space — The metric supplies distances between terms.
• Sequence — The Cauchy obligation quantifies over sequence indices.
• Quantifiers and explicit domains — 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.