Suppose $X$ is locally compact, Hausdorff, and has a countable family of compact subsets whose union is all of $X$. I want to show that $X$ is second-countable. Suppose the family of countable compact subsets are labelled $C_i$, $i \in \Bbb N$, Then there are a couple of ideas that come to mind but I am having trouble putting it all together.
1) Taking the complements of the $C_i$ (problem: what if an element is in all of them, also the complements might not satisfy the requirements of a basis)
2) Using local compactness to say that every point has a compact set containing it with an open neighborhood (also containing it), but how will this relate to the $C_i$ ?
Any hints appreciated.