Let $\Omega$ be an open set in $\mathbb{R}^2$, $K \subseteq \Omega$ and K be compact.
Prove that there exists $r>0$ such that $E=\bigcup_{z \in K} \bar{D}(z,r)$ is a compact subset of $\Omega$, where $\bar{D}(z,r)$ is the closed disc centred at $z$ with radius $r$.
I found Rudin using this fact in one of the proofs in his 'Real and Complex Analysis', however I am having difficult proving this fact is true.
My attempt: Let $2r$ be the minimum distance from the boundary of $\Omega$ to K, then $D(z,r)$ will all lie in $\Omega$. Since the union of $D(z,r)$ covers K, we can reduce the collection into a finite covering of K, say $D(z_1,r),D(z_2,r),...,D(z_N,r)$ Then $\bigcup_{n=1}^{N} \bar{D}(z_n,r)$ will be a compact subset of $\Omega$. However this set is not E, and it is necessary for the result to work for E for the proof to work.
Any help is appreciated!