Compact subset of locally compact $\sigma$-compact Let $X$ be a non-compact, locally compact space. Also suppose that there is a sequence of compact non-empty sets $\{K_n\}_{n\in N}$ such that $$X=\bigcup_{n\in N} K_n,\quad K_n\subset K_{n+1}.$$ 
Now could we say that for any compact subset $K\subset X$ there is a number  $n\in N$ such that $K\subset K_n$? If not, which conditions can help? or, is it possible if we choose sequence of compact sets? How about topological groups?
 A: No, not as stated.
Let $X = [0,1)$ be the half-open unit interval, which is locally compact but not compact.  Let $K_n = \{0\} \cup [\frac{1}{n}, 1-\frac{1}{n}]$ which is compact. Then we have $K_n \subset K_{n+1}$ and $X = \bigcup_n K_n$.  But the compact set $K = [0, \frac{1}{2}]$ is not contained in any of the $K_n$.
However, under these assumptions here is something we can prove.

There exists a sequence of compact sets $K_n'$ such that $\bigcup_n K_n' = X$ and $K_{n-1}' \subset (K_{n}')^\circ$ for each $n$.  In particular, for any compact set $K$, there is some $n$ with $K \subset K_n'$.

Proof.  By local compactness, for each $x \in X$ there is an open set $U_x$ such that $x \in U_x$ and $\overline{U_x}$ is compact.  We construct $K_n'$ recursively.  To get started, let $K_0'=\emptyset$.  Now to construct $K_n'$, suppose $K_{n-1}'$ is already constructed.  Since $K_n \cup K_{n-1}'$ is compact, there exist $x_1, \dots, x_r$ such that $K_n \cup K_{n-1}' \subset U_{x_1} \cup \dots \cup U_{x_r}$.  Set $K_n' = \overline{U_{x_1}} \cup \dots \cup \overline{U_{x_r}}$ which is compact.  Then $K_{n-1}' \cup K_n \subset (K_n')^\circ$.
In particular, $\bigcup_n K_n' \supset \bigcup (K_n')^\circ \supset \bigcup_n K_n = X$.  So the $K_n'$ cover $X$.
Moreover, the sets $(K_n')^\circ$ are an increasing open covering of $X$.  So if $K$ is any compact set, we must have $K \subset (K_n')^\circ \subset K_n'$ for some $n$.
A: I can't comment due to lack of reputation.
In the argument of @Nate Eldredge, do we need the space $X$ to be a Haussdorf space, in order to have closed compact neighbourhoods of each point?
