Consider the following properties of the Hausdorff distance in $\mathbb R^n$.
Let $\Omega_n \supset \Omega_{n+1} \supset ...$ a sequence of open, convex and bounded sets with $\operatorname{int}(\bigcap \Omega_n) \neq \emptyset$. Then $\Omega_n \rightarrow \operatorname{int}(\bigcap \Omega_n)$ in the Hausdorff distance.
If $K_n$ is a sequence of compact sets converging in the Hausdorff distance to a compact set $K$, then for each $x \in \partial K$ exists a sequence $x_n \in \partial K_n$ that converges to $x$.
Times ago I found on the internet the proof of the above properties in a lecture note. But now the site does not exists and I am not finding a book with these properties. Someone could point me a book with the properties above?