Condition for a Metric Space to Be Complete
Theorem
A metric space
is 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 ofthen
is proved here.
Let
be a Cauchy sequence in X.
Define
and so on.
Thenandand all the
are closed, nonempty subsets of
Hence
Let
and take
then there exists
such that for
Hence
For all
hence