Countably Compact Metric Space is Compact/Proof 1

Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $M$ be countably compact.

Then $M$ is compact.

Proof

This follows directly from:

Metric Space is First-Countable

Countably Compact First-Countable Space is Sequentially Compact

Sequentially Compact Metric Space is Second-Countable

Second-Countable Space is Compact iff Countably Compact

$\blacksquare$

Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice, by way of Sequentially Compact Metric Space is Second-Countable.

Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.

As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.