We define the diameter of a nonempty subset A of M by $\displaystyle diam(A)=sup${$\displaystyle d(a,b): a,b \in A$}. Show that A is bounded if and only if diam(A) is finite.

this is seems obvious, if diam(A) is not finite, then the distance from a to b can be infinite, which says that A is not bounded.

However I cannot think of an official proof of this, can anyone help?