# lower bound

Let $$X$$ be an ordered set, and let $$A$$ be a subset of $$X$$. Then an element $$l$$ of $$X$$ is said to be a lower bound of $$A$$ if $$l \leq a$$ for every $$a\in A$$. If in addition $$l \geq m$$ for every lower bound $$m$$ of $$A$$, then $$l$$ is said to be a greatest lower bound or infimum of $$A$$.

The terms upper bound, least upper bound, and supremum are defined analogously.