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Consider the metric space $\bf R$ with the standard Euclidean metric $d$ and let $F(\bf R)$ denote the collection of all finite subsets of $\bf R$. Endow $F(\bf R)$ with the Hausdorff metric $d_H$. See Wikipedia for the definition of $d_H$.

Is $F(\bf R)$ complete w.r.t. $d_H$?

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No. Take a dense subset $\{d_1,d_2,d_3,\ldots\} $ of $[0,1]$. For each $n$, let $F_n=\{d_1,\ldots, d_n\}$. Then $(F_n)$ converges to $[0,1]$ and is therefore a Cauchy-sequence. But the limit $[0,1]$ is obviously not a finite set.

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