Let $\mathcal{K}$ be the collection of non-empty closed bounded subsets of $\mathbb{R}^{n}$.

(a) Show that, if $A, B \in \mathcal{K}$ and we write

$$A+B=\{a+b: a \in A, b \in B\}$$

then $A+B \in \mathcal{K}$.

(b) Show that, if $K_{n} \in \mathcal{K}$, and

$$K_{1} \supseteq K_{2} \supseteq K_{3} \supseteq \cdots$$

then $K:=\bigcap_{n=1}^{\infty} K_{n} \in \mathcal{K}$.

(c) Assuming the result that

$$\rho(A, B)=\sup _{a \in A} \inf _{b \in B}|a-b|+\sup _{b \in B} \inf _{a \in A}|a-b|$$

defines a metric on $\mathcal{K}$ (the Hausdorff metric), show that if $K_{n}$ and $K$ are as in part (b), then $\rho\left(K_{n}, K\right) \rightarrow 0$ as $n \rightarrow \infty$.