How can I show that $y\in R^p$ such that $\lVert y-x\rVert >1/m$ is open when x is on complement of compact subset on $R^p$?

Question

Let $K \subset$ $R^p$ is compact and $x\in R^p$\ $K$.

Then, $G_m$ = {$y\in R^p$ | $\lVert y-x\rVert$$> \frac{1}{m}$ } open

I tried that Suppose $G_m$ is open and then it consists of all interior point.

But I cannot go on....

and also I supposed that $G_m$ is closed and tried to find a contradiction to the property that : It must contain all of boundary points.

But I couldn't...

How can I prove this?