InfC

InfC denotes the infimum (greatest lower bound) of a subset C within a partially ordered set, most commonly the real numbers. It is the largest element that is not greater than every member of C.

Formally, a is the infimum of C if a ≤ c for all c in C, and for every b that is a lower bound of C (b ≤ c for all c in C), we have b ≤ a. In the real numbers, the infimum of C exists whenever C is nonempty and bounded below. If C has a minimum element, then inf C equals that minimum. If C has no minimum but is bounded below, inf C exists and is the greatest lower bound that may not belong to C. If C is unbounded below, inf C does not exist in the real numbers, though in the extended real line the infimum can be defined as −∞.

Examples illustrate the distinction between infimum and minimum. For C = {1, 2, 3}, inf C = 1 and min C = 1. For C = (0, 1], inf C = 0 but C has no minimum since 0 is not an element of C. For C = {1/n : n ∈ N}, inf C = 0, with no minimum.

Infimum is dual to supremum, which is the least upper bound. Together they describe the bounding behavior of sets under a given order. Inf C is widely used in analysis and optimization, where it often represents the best guaranteed lower performance or the limit of a sequence from below.