I am being asked to prove the following for some homework:
Let $K$ be a nonempty compact subset of an open set $U\subset\mathbb C$. Show that there is $r>0$ such that $D(z,r)\subset U$ for any $z\in K$. Note that the $r$ does not depend on $z\in K$.
My possible attempt at a proof and its flaw:
So now that I have what I am trying to prove up on the wall, here is my attempt at a proof and its flaw. In class, we were given the following theorem:
Theorem 1.4.6 Let $A$ and $B$ be nonempty subsets of $C$. Suppose $A$ is compact and $B$ is closed. Then there exist $z_0 ∈ A$ and $w_0 ∈ B$ such that $|z_0 − w_0| = dist(A, B)$. In other words, the minimum of the set {|z − w| : z ∈ A, w ∈ B} exists.
My plan is to use this theorem to show that $dist(K, U^c)$ exists. Then, if I let $r = dist(K, U^c)$, that should show that $D(z,r) \subset U$. The issue is how do I show that $r = dist(K, U^c)>0$. Any advice? Thanks in advance
Theorem 1.4.6 tells you that your $r$ is the distance between some point $z_0\in K$ and some point $w_0\in U^c$. Since $K\subseteq U$, the points $z_0$ and $w_0$ are distinct, so their distance is $>0$.