Here's a rushed proof. you can streamline it.

Assume there is a sequence of points in that converges to . Then lies in any closed set containing . That is, .

Conversely, assume . Then the "ball" of radius centered at will contain points in . For each , pick one of these points and call it , then the sequence converges to

I assume you know how to describe a "ball" in a metric space. we derive this from the metric