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The Class of Non-empty Compact Subsets of a Compact Metric Space is Compact
Let $(M,d)$ be a metric space and let $\mathcal K(M)$ denote the set of all non-empty compact subsets of $M$. This collection is a metric space when equipped with the Hausdorff distance $h$.
I want to prove$(M,d)\mbox{ is compact}\implies(\mathcal K,h)\mbox{ is compact}.$ The statement is true according to the book [V. I. Istratescu, Fixed Point Theory: An Introduction], but the proof is omitted. I have already shown that $M$ is complete implies that $\mathcal K$ is complete.