Heine-Borel Theorem/Metric Space

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Then $M$ is compact iff $M$ is complete and totally bounded.

Necessary Condition
Let $M$ be compact.

Then from Compact Metric Space is Complete, $M$ is complete.

From Compact Metric Space is Totally Bounded, $M$ is totally bounded.

Sufficient Condition
Let $M$ be both complete and totally bounded.

Then from Complete and Totally Bounded Metric Space is Sequentially Compact, $M$ is sequentially compact.

Hence $M$ is compact by Sequentially Compact Metric Space is Compact.

Also see

 * Heine-Borel Theorem
 * Heine-Borel Theorem: Special Case