Theorem
A metric spaceis complete if and only if every nested sequence
of nonempty closed subsets of
with
(diameter tending to 0) has a nonempty intersection so that
Proof
If a complete metric space has every countable nested sequence()
of nonempty subsets of
then
is proved here.
Letbe a Cauchy sequence in X.
Define
and so on.
Thenand
and all the
are closed, nonempty subsets of
Hence
Letand take
then there exists
such that for
Hence
For allhence