Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 2H / c / Solution

Because $K\subseteq K_n$, one of the two directed terms in the given Hausdorff distance vanishes, andSuppose this does not tend to zero. Then some $\epsilon >0$, a strictly increasing sequence $(n_j)$ and points $x_j\in K_{n_j}$ satisfyAll $x_j$ lie in the compact set $K_1$, so after passing to a subsequence, $x_j\to x$. For every fixed $m$, eventually $n_j\ge m$, so $x_j\in K_m$ and the closedness of $K_m$ gives $x\in K_m$. Hence $x\in K$, contradicting $|x_j-x|\to 0$. Therefore $\rho (K_n,K)\to 0$.