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 $F$?

Let $d=\inf\{\text d(a,b):a\in K, b\in \partial A\}$. Let $K_1=\left\{x\in A : \text d(x,K)\le \frac d2\right\}$.  I believe that should work.
