lower bound

A lower bound for a set is a floor: a number that no element of the set ever drops below. Picture the set on a number line; a lower bound is any point sitting at or to the left of every mark.

Formally, a number m is a lower bound for a set S of reals if m is less than or equal to x for every x in S. As with upper bounds, m need not lie in S, and once a set has one lower bound it has infinitely many, since anything smaller works too. A set with at least one lower bound is called bounded below.

Lower bounds are the mirror image of upper bounds, obtained by reflecting through zero: m is a lower bound of S exactly when minus m is an upper bound of the reflected set. The greatest lower bound, when it exists, is the infimum, the counterpart of the supremum.