Connected, locally compact, paracompact Hausdorff space is exhaustible by compacts I'm trying to understand the proof of the following:

A connected, locally compact, paracompact Hausdorff space $X$ has an exhaustion by compact sets,

That is, there exists a sequence $(K_n)_n$ of compact subsets of $X$ whose union is $X$ itself, such that $K_n$ is included in the interior of $K_{n+1}$, for all $n$.
The proof goes as follows. Choose a locally finite open cover $(U_i)_{i∈I}$ of $X$, such that the closure of $U_i$ is compact, for all $i$. Then every compact subset of $X$ intersects only finitely many of the $U_i$ (why?).
Then $K_1$ is chosen as the closure of any nonempty $U_i$. $K_2$ is chosen as the union of the closures of the $U_i$ that intersect $K_1$ and so on. The rest is easy.

Suppose $K\subseteq X$ is compact.  For each $x\in K$, choose an open set $V_x$ around $x$ that intersects only finitely many $U_i$ (by local finiteness).  Since $K$ is compact, it is covered by only finitely many of these $V_x$.  Since each $V_x$ intersects only finitely many $U_i$, so does $K$.
