Let $\mathcal{K}$ be the set of all non-empty compact subsets of $m$-dimensional Euclidean space $\mathbb{R}^{m}$. Define the Hausdorff metric on $\mathcal{K}$, and prove that it is a metric.

Let $K_{1} \supseteq K_{2} \supseteq \ldots$ be a sequence in $\mathcal{K}$. Show that $K=\bigcap_{n=1}^{\infty} K_{n}$ is also in $\mathcal{K}$ and that $K_{n} \rightarrow K$ as $n \rightarrow \infty$ in the Hausdorff metric.