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Condition for a Metric Space to Be Complete

Theorem

A metric spaceis complete if and only if every nested sequenceof nonempty closed subsets ofwith(diameter tending to 0) has a nonempty intersection so that

Proof

If a complete metric space has every countable nested sequence()of nonempty subsets ofthenis proved here.

Letbe a Cauchy sequence in X.

Define

and so on.

Thenandand all theare closed, nonempty subsets ofHence

Letand takethen there existssuch that for

Hence

For allhence