**Question.** *Let $V_1,\ldots,V_n$ be open, bounded and convex subsets of $\mathbb R^2$. Show that $F=\mathbb R^2\smallsetminus\bigcup_{i=1}^n V_i$ possesses only finitely many connected components.*

I have managed to produced a rather long proof in the case when the $V_i$'s are rectangles, with sides parallel to the axes. As this looks intuitively almost obvious, I am wondering whether there is some straight-forward way to show this.