Conway Exercise $8.3.1$ Let $K$ be a compact subset of the open set $G$ and suppose that any bounded component $D$ of $G-K$ has $cl(D)\cap \partial G\neq \varnothing$. Then, every component of $\mathbb{C}_\infty - K$ contains a component of $\mathbb{C}_\infty- G$
I'm having trouble solving this problem. My idea was to try to prove it by the contrapositive: suppose a component $A$ that doesn't. Then we have two cases $\{\infty\}\in A$ and $\{\infty\}\neq A$ ($A$ bounded). The second case was better to envolve, but I don't know how to proceed with the first one.
I'd appreciate any hint about this problem.