infR

infR is a notation sometimes used in mathematical writing and software to denote the infimum, or greatest lower bound, of a set R. In the standard setting, R is a nonempty subset of a totally ordered set, most commonly the real numbers. The infimum of R, written inf R or infimum of R, is the largest value that is less than or equal to every element of R. Existence requires that R be bounded below; when R is bounded below, the infimum exists in the extended real line, and if R is bounded and the infimum happens to belong to R, it is the minimum of R.

Examples illustrate the concept: inf {3, -2, 4} = -2, since -2 is a lower bound for the set and no larger number can serve as a lower bound. inf {1/n : n in N} equals 0, though 0 is not an element of the set. For the open interval (0,1), the infimum is 0. If R = {x in R : x ≥ 2}, the infimum is 2, which is also the minimum of R.

In broader contexts, infimum is a fundamental notion in order theory and analysis. It is dual to the supremum (least upper bound) and extends to general partially ordered sets where the greatest lower bound may or may not exist. When used in texts, inf R is often interchangeable with “infimum of R,” though some authors prefer spacing (inf R) to avoid ambiguity with a variable or function name such as infR in code.

Note that infimum should not be confused with the limit inferior (lim inf) of a sequence, a related but distinct concept in analysis.