= Solution
Because $K\subseteq K_n$, one of the two directed terms in the given <Hausdorff distance> vanishes, and
$$
\rho(K_n,K)=\sup_{x\in K_n}\inf_{y\in K}|x-y|.
$$
Suppose this does not tend to zero. Then some $\varepsilon>0$, a strictly increasing sequence $(n_j)$ and points $x_j\in K_{n_j}$ satisfy
$$
\inf_{y\in K}|x_j-y|\geq\varepsilon.
$$
All $x_j$ lie in the compact set $K_1$, so after passing to a subsequence, $x_j\to x$. For every fixed $m$, eventually $n_j\geq m$, so $x_j\in K_m$ and the closedness of $K_m$ gives $x\in K_m$. Hence $x\in K$, contradicting $|x_j-x|\to0$. Therefore $\rho(K_n,K)\to0$.
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