Proof That a Sequentially Compact Subset of a Metric Space is Compact
A sequentially compact subsetof a metric spaceis compact.
Letbe a sequentially compact subset of a metric spaceand letbe an open cover ofThe sethas a Lebesgue numbersince 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 subsetswith whereis the diameter of the subset
is a Lebesgue number ofso there are open setssuch that
Henceandis a finite subcover. Henceis compact.