Showing $\rho$ is a metric inside a metric space $(X, d)$

Question

Let $(X, d)$ be a metric space and $B(X)$ be the family of closed, bounded, nonempty subsets of $X$. For $A, B ∈ B(X)$ let

$$ρ(A, B) := \max\{\sup_{b \in B} d(b, A),\ \sup_{a \in A} d(a, B)\},$$

where $d(x, A) = \inf\{d(x, a) : a ∈ A\}$ for $x ∈ X$ and $A ⊂ X$.

How can I show that $ρ$ is a metric on $B(X)$?