Show that compact set is contained in the interior of another compact set

Question

Problem: Let $U \subset \mathbb{R^n}$ be open. Let $C \subset U $ be compact. Show that there exists a compact set $D \subset U$ such that $C \subset \text{int}(D)$.

Please give some hint, I could not make any progress.

Answer 1

This is true in a locally compact space $X$ like $\mathbb{R}^n$.

For every $x \in C$ pick $U_x$ open such that $\overline{U_x}$ is compact. (In the Euclidean spaces, just take an open ball of some size). Finitely many $U_x$ cover $C$ by compactness. The union of their closures is $D$. Its interior contains those $U_x$ that cover $C$.