supremum

Supremum is a fundamental concept in mathematics, particularly in the field of real analysis and order theory. It refers to the least upper bound of a set within a given ordered set. Specifically, if a set S is bounded above—meaning there exists some element greater than or equal to every element in S—the supremum of S is the smallest such element that still holds this property.

For example, consider the set S = {x ∈ R | x < 3}. Although 3 is not an element of S, it is an upper bound because every element of S is less than 3. The supremum of S is 3 because it is the smallest upper bound, even though it is not contained within S.

The concept of supremum is essential in defining limits, continuity, and optimization problems. In real numbers, the supremum of a set, if it exists, is unique. It may or may not be an element of the set itself; when it is part of the set, it is also called the maximum. The existence of a supremum is guaranteed for any non-empty set that is bounded above in the real numbers, according to the completeness property of the real number system.

Supremum is also known as the least upper bound and is denoted by contrary symbols such as "sup" to indicate the operation. Its dual concept, infimum, refers to the greatest lower bound of a set, or the largest element that is less than or equal to all elements in the set. Both concepts are pivotal for rigorous mathematical analysis and theoretical foundations.