Let be a Sequentially compact Metric space with open cover . We can find a s.t. for any there is a s.t. .

Proof

Suppose there is no such . For any find s.t. there is no s.t. . As is sequentially compact, find a convergent subsequence where . Suppose it converges to . Then we find an open which contains . Hence, there is an open ball for . But for large enough , certainly which is a contradiction.