This is a problem from Munkres' Topology.
Let $X$ be a metric space.
(a) Suppose that for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete.
(b) Suppose that for each $x \in X$, there is an $\epsilon \gt 0$ such that the ball $B(x, \epsilon)$ has compact closure. Show by means of an example that $X$ need not be complete.
I'm currently stuck on (a), I'm trying to show that given a Cauchy sequence ($x_n$), some tail of the sequence belongs entirely on a ball $B(x,\epsilon)$ and thus since a compact set is sequentially compact in metric space, the sequence has a convergent subsequence and so the Caucy sequence converges to the same limit. However, I'm having trouble constructing this process. Also, for (b), I cannot think of any such example. Any solution, hint, or suggestions would be appreciated.