Let $K \subseteq U\subseteq R^{n} $, where K is compact, nonempty and U is open.
Let $\epsilon = d(K, \partial U)/2$. Show that closure of set $V = B_{\epsilon}(K) $ is compact and $K \subseteq V\subseteq \bar{V}\subseteq U $.
My thoughts: I have no problem showing closure of $V$ is bounded. Closure is closed by definition, therefore closure of V is compact. $K \subseteq V$ is true, because if $x\in K$ then $d(x,K)=0$ so $d(x,K)< \epsilon$. But I can't show the last part: $\bar{V}\subseteq U $.