Accidentally Invoking a Higher Power

I accidentally invoked the Axiom of Countable Choice in my real analysis assignment. Oops. Kind of.

The problem, in case anyone is interested: prove that if a metric space (X, d) has a countable basis of open sets, then it is separable.

Incidentally, the converse is true also: a separable metric space has a countable basis of open sets.

Oh well, most mathematicians are pro-choice anyway. Yes, I believe that every vector space has a basis, even if I don't understand the intricacies of the matter (Banach-Tarski paradox and all that).