 Solution

A subset $B$ of a metric space is bounded if it lies in a ball of finite radius: there exist a point $x_0$ and $R<\infty$ such that $d(x,x_0)\le R$ for every $x\in B$. Equivalently its diameter ${\sup }_{x,y\in B}d(x,y)$ is finite, with the empty set bounded by convention.