Sequentially Compact Metric Space is Separable

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Theorem

A sequentially compact metric space is separable.


Proof

This follows directly from:

$\blacksquare$


Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice, by way of Totally Bounded Metric Space is Separable.

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.