Hausdorff distance

Let $(X,d)$ be a metric space, then denote by $\mathcal{F}_X$ the family of all closed and bounded subsets of $X$. Given $A\in \mathcal{F}_X$, let denote by $N_r(A)$ the neighborhood of $A$ of radius $r$: $$\cup_{x\in A} B(x,r).$$

Then the Hausdorff distance between $A,B\in \mathcal{F}_X$ is given by $$\delta(A,B) = \max\{\inf \{r>0 : B\subset N_r(A)\},\inf \{r>0 : A\subset N_r(B)\}\}.$$

Theorem. $\delta$ is a metric.

Theorem. If $(X,d)$ is complete, then so is $(\mathcal{F}_X,\delta)$.