Let $\Omega \subset \mathbb{R}^n$ be an open set. Let $K\subset\Omega$ be compact. Then show that there exists an $r>0$ such that the set
$\{y\in\mathbb{R}^n : ||y-x||\leq r$ for some $x$ in $K\}$
is compact in $\Omega$.
Any direction/help would be useful. Thank you.
If $r$ is small enough the the set is contained in $\Omega$: this is because the distance between the compact set $K$ and the closed set $\Omega^{c} $ is positive. We can take any $ r \in (0, d(K,\Omega^{c})$. Once we choose $r$ to make this a subset of $\Omega$ we just observe that the set is closed and bounded, hence compact.