I want to show that the metric space $(I([0,1]), d_H)$ is complete, where $I([0,1])$ is the set of closed intervals $[a,b]$, $0 \leq a \leq b \leq 1$, and $d_H$ is the Hausdorff distance.
I've tried to prove that the limit of every Cauchy sequence is in $I([0,1])$. Suppose we have a Cauchy sequence {$[a_k, b_k]$}, which induces the bounded sequences {$a_k$}, {$b_k$}. By Bolzano-Weierstrass, these two have convergent subsequences {$a_{n_k}$}, {$b_{n_k}$}, with limits
$$\lim_{k \to \infty} a_{n_k} = a,$$
$$\lim_{k \to \infty} b_{n_k} = b.$$
And I guess I want to show that that $\lim_{k \to \infty} [a_k, b_k] = [a,b]$, but I'm unsure how to proceed.
Any help would be highly appreciated.
Best Answer
I saw your question a while ago, but I didn't know how to answer it. As time have passed and nobody answered you, I'll write my approach, which I don't know if it's correct.
Let $\{[a_n,b_n]\}_{n\geq 1}\subset I([0,1])$ be a Cauchy sequence. Then, for every $\varepsilon>0$ exists $n_0\in\mathbb{N}$ such that $d_H([a_n,b_n],[a_m,b_m])<\varepsilon$, for every $n,m>n_0$. Now, for intervals, the following relation is verified: \begin{equation*} d_H([a,b],[c,d])=\max\{|a-c|,|b-d|\}. \end{equation*}
Then, in our case, for every $\varepsilon>0$ exists $n_0\in\mathbb{N}$ such that $\max\{|a_n-a_m|,|b_n-b_m|\}<\varepsilon$, for every $n,m>n_0$. In other words, $|a_n-a_m|<\varepsilon$ and $|b_n-b_m|<\varepsilon$, for every $n,m>n_0$. With this, we observe that $\{a_n\}_{n\geq 1}$ and $\{b_n\}_{n\geq 1}$ are both Cauchy sequences of real numbers. Since $(\mathbb{R},|\cdot|)$ is a complete space, there exist $a,b\in\mathbb{R}$ such that $a_n\rightarrow a$ and $b_n\rightarrow b$.
To end, let's see that $[a_n,b_n]\rightarrow [a,b]$, in the sense of the Hausdorff distance. Using the same property as before we have that \begin{equation*} d_H([a_n,b_n],[a,b])=\max\{|a_n-a|,|b_n-b|\}\rightarrow 0, \end{equation*} since $a_n\rightarrow a$ and $b_n\rightarrow b$. So, the Cauchy sequence $\{[a_n,b_n]\}_{n\geq 1}$ is a convergent sequence, which finish the proof.
I hope you find this useful and, if you spot any mistake in the reasoning, please comment it.