least upper bound

A number is an upper bound of a set S of numbers if every member of the set is less than or equal to it. A number n is the least upper bound of the set if there is no number less than it that is also an upper bound. It is a non-trivial result that every set of real numbers that has an upper bound has a least upper bound. If we reverse the order, considering ‘greater than or equal to’ instead of ‘less than or equal to’, we obtain the mirror notion of a greatest lower bound.