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?

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?
