Containing a net with a compact set.

Question

Let $G$ be a locally compact group and let $(x_{\alpha})$ be a convergent net, say to $x$, in $G$. Is it possible to construct a compact subset $K$ of $G$ which contains each $x_{\alpha}$ and $x$?

Answer 1

Not necessarily. Consider $G=\Bbb R,$ $x=0,$ and the net $(0,\infty)\to\Bbb R$--with $(0,\infty)$ directed in increasing order--given by $x_\alpha=\frac1\alpha.$ Any subset of $\Bbb R$ containing every $x_\alpha$ must fail to be bounded, so fail to be compact.