How to prove that a second-countable, locally compact Hausdorff space admits an exhaustion by compact sets? I'm trying to understand one step in a proof that I've seen.  X has a basis of precompact open subsets.  Let $(U_i)_{i=1}^\infty$ be a countable cover.  Let $K_1=\overline{U}_1$ and.  Assume by induction that we have built sets $K_1,...,K_k$ such that $U_j\subseteq K_j$ for each $j$ and $K_{j-1} \subseteq$ Int $K_j$ for $j \geq 2$.  Since $K_k$ is compact, there is some $m_k$ such that we have $K_k \subseteq U_1 \cup...\cup U_{m_k}$.  We then let $K_{k+1} = \overline U_1 \cup...\cup \overline U_{m_k}$.  So, $K_{k+1}$ is compact and $K_k \subseteq$ Int $ K_{k+1}$.  If necessary, we may increase $m_k$ so that $m_k \geq  k+1$ so that $U_{k+1} \subseteq K_{k+1}$.  We then finish the proof with induction.
Here's what I don't understand.  When would we ever have to increase $m_k$?  The only cases I can imagine are ones where, e.g., the first few $U$s are the empty set.
Lee gives a proof of this in his smooth manifolds book and another in his topological manifolds book.  And in one he increases $m_k$ so that it is greater than $k+1$ and in the other greater than or equal to $k+1$ making me wonder whether he even knows exactly what cases he is imagining.
Another thought that I had was that it is really just a residue of wanting to have the induction step isolated from any richer context.
A final thought that I had was that maybe I'm just off by a 1 somewhere in the index.
So, are there any interesting non-trivial examples where we would actually need to increase $m_k$?
 A: Here's an example in which you need to increase $m_k$. Let $X$ be the union of a countably infinite collection of disjoint circles in the plane, and for each $j$, let $U_j$ be one of the circles. Then in the proof, $K_1 = \overline U_1 = U_1$, and if you don't make sure $m_1$ is at least $2$, then you might end up with $K_2 = \overline U_1$, and you wouldn't have $U_2\subseteq K_2$. (Of course, a countably infinite set with the discrete topology would illustrate the point just as well, but the circles are what I thought of first.)
There's no good reason why I insisted on $m_k> k+1$ in Introduction to Topological Manifolds and only $m_k\ge k+1$ in Introduction to Smooth Manifolds. My guess is that when I wrote the first book, I tossed in $m_k>k+1$ out of an abundance of caution, and then when I got around to writing the second book, I realized that it was perfectly sufficient to write $m_k\ge k+1$. The proof works fine either way, so there was no reason to go back and post a correction to the first proof.
