Theorem
A sequentially compact subsetof a metric space
is compact.
Proof
Letbe a sequentially compact subset of a metric space
and let
be an open cover of
The set
has a Lebesgue number
since every open cover of a sequentially compact subset of a metric space has a Lebesgue number. Since
is totally bounded, there is a decomposition of
into a finite number of subsets
with
where
is the diameter of the subset
is a Lebesgue number of
so there are open sets
such that
Henceand
is a finite subcover. Hence
is compact.