The problem says:
If every closed ball in a metric space $X$ is compact, show that $X$ is separable.
I'm trying to use an equivalence in metric spaces that tells us: let X be the matric space, the following are equivalent
X is 2nd countable
X is Lindeloff
X is separable
I also thought about taking balls of a "big" radius and that they are disjoint, but I do not see how to make the set of balls is countable.