Proof That a Sequentially Compact Subset of a Metric Space is Compact
Theorem
A sequentially compact subset
of a metric space
is compact.
Proof
Letbe a sequentially compact subset of a metric spaceand let
be an open cover of
The sethas a Lebesgue number
since every open cover of a sequentially compact subset of a metric space has a Lebesgue number. Sinceis totally bounded, there is a decomposition ofinto a finite number of subsets
with 
where
is the diameter of the subset
is a Lebesgue number ofso there are open sets
such that
Hence
and
is a finite subcover. Henceis compact.