Condition for a Metric Space to Be Complete


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


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

Letbe a Cauchy sequence in X.


and so on.

Thenandand all theare closed, nonempty subsets ofHence

Letand takethen there existssuch that for


For allhence

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