or a member of greater than (if ). Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals. There is a corresponding ; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set. If in a partially ordered set every bounded subset has a supremum, this applies also, for any set in the function space containing all functions from to where if and only if for all For example, it applies for real functions, and, since these can be considered special cases of functions, for real -tuples and sequences of real numbers. The least-upper-bound property is an indicator of the suprema. Infima and suprema of real numbers In analysis, infima and suprema of subsets of the real numbers are particularly important. For instance, the negative real numbers do not have a greatest element, and their supremum is (which is not a negative real number). The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset of the real numbers has an infimum and a supremum. If is not bounded below, one often formally writes If is empty, one writes Properties If is any set of real numbers then if and only if and otherwise If are sets of real numbers then (unless ) and Identifying infima and suprema If the infimum of exists (that is, is a real number) and if is any real number then if and only if is a lower bound and for every there is an with Similarly, if is a real number and if is any real number then if and only if is an upper bound and if for every there is an with Relation to limits of sequences If is any non-empty set of real numbers then there always exists a non-decreasing sequence in such that Similarly, there will exist a (possibly different) non-increasing sequence in such that Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from topology that if is a continuous function and is a sequence of points in