real analysis – Bounded component of the complement of a compact set

Suppose $$A_1$$ and $$A_2$$ are two compact sets in $$mathbb{R}^n$$ for $$n>1$$. Suppose these sets are disjoint. It seems correct that if the complement of their union has a bounded component $$C$$, and $$overline{C}cap A_1not=emptyset$$, then $$C$$ must be the bounded component of $$A_1$$. Is this correct? How to prove it?