Let $\Omega_1 \supset \Omega_2 \supset ...$ a sequence of nonempty , open, bounded and convex sets.

Define $\Omega = int \Bigl( \overline{\displaystyle\bigcap_{k=1}^{\infty} \Omega_k } \ \Bigl) $ and supoose that $\Omega$ is nonempty, open, convex and bounded.

i believe that $\partial \Omega_k \rightarrow \partial \Omega$ in the Hausdorff distance. In a article that i am studying appears the author is using this fact. I'm not finding this result in the literature, but probably is a basic result of topology. Someone can say to me a book or a article with a proof or just saying that the affirmation is true?