Suppose (for a contradiction) that for all there exists such that contains only a finite number of terms of a given sequence then since is compact, the cover has a finite subcover ie. there exists such that but this is clearly a contradiction since each of these balls contain only finitely many terms. So we conclude that for any given sequence there exists a such that for all contains infinitely many terms of said sequence ie. there exists a subsequence converging to . This in turn proves that is closed trivially.