I feel there's something wrong with my solution, so I'd like to check it tout.
Let X be a metric space and suppose that there exists some ε > 0 such that every ε-ball in X has a compact closure. Show that X is complete.
(btw, it would be enough, in the formulation of the problem to assume that every ε-ball in X is compact, since X is Hausdorff, and compact subspaces are closed, right?)
The Attempt at a Solution
So, let xn be a Cauchy sequence in X. Choose ε > 0 such that every ε-ball has a compact closure. Choose N such that for any n, m >= N, we have |xn - xm| < ε. Hence, there exists an ε-ball containing all but finitely many members of the sequence xn. For this finite number of members of the sequence, choose for each an ε- ball containing it. Now, a finite union of such compact ε-balls is compact, and hence sequentially compact. So, xn has a convergent subsequence, so X is complete. (actually, more precisely, the union of the ε-balls is complete, but since they are a subspace of X, and we have chosen a Cauchy sequence in X, we arrived at a convergent subsequence in X again, hence I concluded that X is complete=
Thanks in advance, but for some reason, after dealing with more abstract issues, I feel a but unsure about metric spaces.