subsets of the real numbers, another kind of duality holds: where Examples Infima The infimum of the set of numbers is The number is a lower bound, but not the greatest lower bound, and hence not the infimum. More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the minimum of the set. If is a decreasing sequence with limit then Suprema The supremum of the set of numbers is The number is an upper bound, but it is not the least upper bound, and hence is not the supremum. In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete. One basic property of the supremum is for any functionals and The supremum of a subset of where denotes "divides", is the lowest common multiple of the elements of The supremum of a set containing subsets of some set is the union of the subsets when considering the partially ordered set , where is the power set of and is subset. See also (infimum limit) Notes References External links Order theory