Heine-Borel Theorem/Metric Space

Theorem
A metric space is compact it is both complete and totally bounded.

Necessary Condition
This follows directly from:
 * Compact Metric Space is Complete
 * Compact Metric Space is Totally Bounded

Sufficient Condition
This follows directly from:
 * Complete and Totally Bounded Metric Space is Sequentially Compact
 * Sequentially Compact Metric Space is Compact

Also see

 * Heine-Borel Theorem
 * Heine-Borel Theorem: Real Line