This question comes from Ratcliffe's Foundations of Hyperbolic Manifolds.
Let $X$ be a finitely compact metric space (i.e. all closed metric balls are compact). Prove that a discrete group $\Gamma$ of isometries of a $X$ is countable.
$X$ being finitely compact tells us that $\Gamma$ also acts discontinuously on $X$ (this is a theorem in the text). Apparently it also tells us that $X$ is second-countable (the author just says this; I've not seen it proven, but I'm willing to accept is as fact for now). My thought is to somehow appeal to this discontinuous action on the closed, compact metric balls $\bar{B_i}$, where $\{B_i\}$ is the countable base for the topology. However, I've not successfully figured out a way to mix these two ideas.
Does anyone have any suggestions for other approaches?
If it matters, this isn't a homework question - I'm studying for my comprehensive exams next month and I thought this seemed like a reasonable practice exercise.
Pick a base point $p$. For each integer valued radius $n$, ask yourself: how many $\gamma \in \Gamma$ are there such that $d(p,\gamma \cdot p) \le n$? Can you see how to use compactness of $B(p,n)$ to prove this number is finite? Can you then finish the proof?