$D=\{z\in\mathbb{C}:|z|<1\}$. Every compact subset $K$ of D, $K \subseteq D$ is in $rD$ for sufficiently large $0\le r<1$

Question

$D=\{z\in\mathbb{C}:|z|<1\}$

How can I show that every compact subset $K$ of D, $K \subseteq D$ is in $rD$ for sufficiently large $0\le r<1$?

When drawing a sketch of this problem it seems clear but how can I prove this?

Answer 1

Consider the map$$\begin{array}{rccc}\delta\colon&K&\longrightarrow&[0,\infty)\\&z&\mapsto&\lvert z\rvert.\end{array}$$Since $\delta$ is continuous and $K$ is compact, it has a maximum. Call it $r^\star$. Take $r\in(r^\star,1)$. Then$$(\forall z\in K):\lvert z\rvert<r.$$In other words, $K\subset D(0,r)=rD$.