A set S of real numbers is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.
While in a metric space a non-empty subset $S$ of metric space $X$ is said to be bounded set if its diameter is finite.
My question is: are both the definitions of boundedness are equivalent? Is it possible to determine diameter of every subset?
Is there any other criterion also to determine whether given subset of a metric space is bounded or not?