Need help verifying/completing proof

Here's a definition before the question

Separable: A metric space is separable if it has a countable dense subset

Base: A collection

of open sets of metric space

is called a base for

if the following is true: For every

and every open set

such that

, we have

Question

Prove every separable metric space has a countable base

Proof

Let

be a countable dense subset of metric space

and let

be a base for

Let

be a neighborhood of

:

,

dense, so either

(in which case

countable, so proof follows) or

are limit points of

So

, for some

and

There are a countable number of sets

, so the base is countable

QED