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?