## Proof That a Finite Subset of a Topological Space is Sequentially Compact

Theorem

A finite subsetof a topological spaceis sequentially compact.

Proof

A subsetof a topological spaceis sequentially compact if every sequence in contains a subsequence which converges to a point in

Letbe a finite subset ofand letbe a sequence in

Sinceis finite, at lease one of the elements insay x_0 , occurs an infinite number of times, We can choose a subsequence consisting of the termswhich converges to