The forward direction follows from Compact subsets of a Hausdorff space are closed and Compact sets in a metric space are bounded.

For the converse, let be closed and bounded. Then it can be enclosed with an -box . Since Closed subsets of a compact space are compact, it is enough to prove is compact.

Suppose is not compact. Then there exists an open cover with no finite subcover. can be broken into sub-boxes of half its side length, at least one of which must require an infinite subcover of . Call this . Continuing this argument iteratively, one obtains a sequence of shrinking -boxes

each requiring infinite subcovers, where has side length . One may construct a sequence such that , which is clearly a Cauchy sequence and thus converges to some by completeness of . By sequential closedness for all . Now since is a cover there exists some such that , and by openness there exists open ball . For sufficiently large , , whence is a finite subcover of , a contradiction. Therefore is compact, so is compact.