infimum

The infimum of a set is its tightest possible floor. Among all numbers that bound the set from below, the infimum is the largest one, the floor you cannot raise even slightly without leaving some element beneath it.

Formally, a number t is the infimum of S, written inf S, if t is a lower bound of S and no number larger than t is a lower bound. Equivalently, for every positive epsilon there is an element of S smaller than t plus epsilon, so the infimum is approached arbitrarily closely from above by members of the set.

The infimum is the exact mirror of the supremum through zero: inf S equals minus the supremum of the set of negatives of elements of S. Because the reals have the least upper bound property, they automatically have the analogous greatest lower bound property, so every nonempty set bounded below has an infimum.

Like the supremum, the infimum may or may not belong to the set. When it does, it coincides with the minimum. For the set of reciprocals 1, 1/2, 1/3, and so on, the infimum is 0, which is not a member.