Let $D$ be a domain (an open, connected set) and suppose $K \subset D$ is compact. Now, define for $\eta > 0$ "small enough" the following set:
$K_{\eta}=\{z \in D: \mathrm{dist}(z,K) \leq \eta \}$
Where it is understood that $\mathrm{dist}(z,K)=\inf\{|z-k|:k \in K\}$. Show that the set $K_{\eta}$ is compact.
I know for a fact that we can show that for every sequence $(z_{n})\subset K_{\eta}$ we must show the existance of a convergent subsequence $(z_{n_{k}})\subset (z_{n})$ whose limit $z$ belongs to $K_{\eta}$, which shows that $K_{\eta}$ is a compact subset. I was thinking of trying to prove this by contradiction, but I've been trying to solve this for a while and I haven't got anywhere. As well, it is clear that $K \subset K_{\eta}$, and $K$ is closed and bounded by compactness, but I'm not so sure if these facts are useful for hte proof.
Any help is appreciated, thanks!