Consider a discrete subgroup $G$ of $R^n$. Then let we have an open ball $B\subset G$ (topology in $G$ is induced from $R^n$). Why is $|B|<\infty$?
For example I can take $A=\{\tfrac{1}{1},\tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4},\cdots, -1\}$, which is discrete set. Naturally I can find an open ball in A which is not finite, however $A$ is not a group. So I guess being group is necessary.