Let $(X,d)$ be a metric space and let $K(X)$ denote the set of all compact subsets of $X$. Then $(K(X), d_H)$ is a metric space, where $d_H$ is the Hausdorff metric.
How can I show that if $X$ if complete, that $K(X)$ is complete?
My try: Start with an arbitrary Cauchy Sequence $\{K_n\}$ in $X$. Then for all $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $d_H(K_n,K_m) < \epsilon$. It has been suggested to me to let $K = \{x : x = \lim_{n \to \infty} x_n \text{ with } x_n \in K_n\}$, and to show that $K_n \to K$. By the triangle inequality $d(K_n,K) \leq d(K_n,K_m) + d(K_m,K_N) + d(K_N,K).$
How to show that $d(K_N,K)$ can be bounded by $\epsilon$?
