Metric Spaces and closed Sets
It's not obvious that the intersection of a decreasing (##B_1\supset B_2\supset\dots##) sequence of open balls is not open. You should think of simple set that you know is not open, and try to express that as a countable intersection of open sets. (The idea to use a decreasing sequence of open balls will work, but it's not your only option).
I use the notation B(x,r) for the open ball around x with radius r. This notation might be useful here.