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
Letbe a Cauchy sequence in X.
and so on.
Thenandand all theare closed, nonempty subsets ofHence
Letand takethen there existssuch that for