Let $F$ Be the non-empty finite bounded subsets of $\mathbb{R}^2$. We want to know if $F$ is complete with respect to the Hausdorff metric.
I made a subset of all natural numbers in the interval $[0,10+1/n]$ this a subset of $F$, and we can construct the Cauchy sequence $G_n$ which converges (according to the Hausdorff metric) to the union of $B(x,\varepsilon)$ with $x \in $$\{0,1,2,..,10\}$ which is not a finite subset of $\mathbb{R}^2$ and thus is $(F,h)$ not complete.
Is this correct? I do not know what I can do to complete this space because I find it very difficult to construct a row with the finite subsets if it can converge to a subset which is uncountable. Can anybody tell me if my first thought are correct and help me to complete this space?
Thanks in advance.
It seems like you have realized that finite sets can converge to infinite sets in the Hausdorff metric. The natural question is to see how far this can be taken. Let $C \subset \mathbb{R}^2$ be a compact set. For each $p \in C$ construct the epsilon ball $B(\epsilon, p)$ centered at $p$. The collection of these balls cover $C$ and we can choose a finite subcover $\{B(\epsilon, p_1), \ldots, B(\epsilon, p_n)\}$. The set $\{p_1, \ldots, p_n\}$ is a finite set whose Hausdorff distance is within $\epsilon$ of $C$. By letting $\epsilon \to 0$ we obtain a sequence of finite subsets of $\mathbb{R}^2$ converging to $C$.
This suggests that the compact subsets of $\mathbb{R}^2$ with the Hausdorff distance is a natural canidate for the space you are looking for. We only have to prove that it is complete. I have seen this theorem proved in section 1.5 of "Geometric Integration Theory" by Parks and Krantz. The book also discusses a lot of other useful properties of this metric space. The book is available publicly online at https://www.math.wustl.edu/~sk/books/root.pdf