Every separable metric space has a countable base

I'd fill in the details of "it's easy to check" steps. They are crucial here (you must use the triangle inequality somewhere in the proof, as the theorem does not hold for non-metric spaces).

Also state what the supposed countable base is, beforehand. Then show it is indeed countable, and then do this proof to show it's actually a base.

The proof itself is correct in essentials. But fill in all the dots.