Consider a set m. If there is an element S, so that no element in M exceeds S, then S is an upper bound of M. If there is a minimum upper bound among all these upper bounds, it is called the supremum of M. A bounded set may have countless upper and lower bounds, but if there is an upper supremum, there can only be one upper supremum.

A necessary and sufficient condition for element A to be supremum: element A is the upper bound of all elements in the specified set B, and all other upper bounds of set B are greater than a (or the set has no other upper bounds).