Condition for a Metric Space to Be Complete
A metric spaceiscomplete if and only if every nested sequenceofnonempty closed subsets ofwith(diametertending to 0) has a nonempty intersection so that
If a complete metric space has every countable nested sequence()ofnonempty subsets ofthenisproved here.
Letbea Cauchy sequence in X.
and so on.
Thenandandall theareclosed, nonempty subsets ofHence
Letandtakethenthere existssuchthat for