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Proof That Compact Metric Spaces Are Complete

Theorem

A compact metric spaceis complete.

Proof

Supposeis a countable family of closed, nonempty subsets of X such that

Sinceis compact,

Letbe a Cauchy sequence in X.

Define

and so on.

Thenandand all theare closed, nonempty subsets of Hence

Letand takethen there existssuch that for

Hence

For allhence

Hence X is complete.