This is Problem 4-29 from John Lee's Introduction to Topological Manifolds.
Let $X$ be a locally compact Hausdorff or a complete metric space. Show that every nonempty countable closed subset of $X$ contains at least one isolated point.
I think I need to use the Baire Category Theorem, which states that for such $X$, every countable collection of dense open subsets has a dense intersection. But I am not sure how to use this to prove the statement. I would greatly appreciate any help.