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

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

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