Condition for a Metric Space to Be Complete

Theorem

A metric spaceiscomplete if and only if every nested sequenceofnonempty closed subsets ofwith(diametertending to 0) has a nonempty intersection so that

Proof

If a complete metric space has every countable nested sequence()ofnonempty subsets ofthenisproved here.

Letbea Cauchy sequence in X.

Define

and so on.

Thenandandall theareclosed, nonempty subsets ofHence

Letandtakethenthere existssuchthat for

Hence

For allhence