Countably Compact Metric Space is Sequentially Compact

Theorem

Let $M$ be a countably compact metric space.

Then $M$ is sequentially compact.

Proof

This follows directly from the results:

Metric Space is First-Countable

Countably Compact First-Countable Space is Sequentially Compact

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces