How to explain this formally? This is an exercise of my homework:


Let $K\subset A\subset \mathbb{R}^N$ such that $N\in\mathbb{N}^*$, $K$ is compact and $A$ is open. Show that there is an $K_1$ compact such that $K\subset {K_1}^o \subset K_1 \subset A$ 


(Where ${K_1}^o$ is the set of interior points of $K_1$)
My strategy is show that there is a closed set $F$ in $A$ containing $K$, but how to do this? In $\mathbb{R}$ this is simple, but how to do in $\mathbb{R}^N$? The idea is that because A is open and K is compact we have (interior) points of A outside of K but how to use this to find $F$?

HINT: For each $x\in K$ there is an $\epsilon_x>0$ such that $B(x,\epsilon_x)\subseteq\operatorname{cl}B(x,\epsilon_x)\subseteq A$. Each of the sets $\operatorname{cl}B(x,\epsilon_x)$ is closed and bounded in $\Bbb R^N$, so it’s compact. Let $\mathscr{U}=\{B(x,\epsilon_x):x\in K\}$; $\mathscr{U}$ is an open cover of $K$. Now use the fact that $K$ is compact.
Here $B(x,\epsilon)$ is the open ball of radius $\epsilon$ centred at $x$ in the usual Euclidean metric.
