Let (X,d) be a metric space. X is said to be separable if there is a countable set Q contained in X such that Q is dense in X. Show that a separable metric space has a basis for its open sets consisting of a countable family of open sets.

I started by establishing the open ball B(q,1/n) where n = 1,2,...

I know I want to show that every open set is a union of these sets.

I'm not sure where to go from here. Any help would be great!