Lower bound

A lower bound of a subset $S$ of an ordered set $(X,\le)$ is an element $\ell\in X$ such that

Lower bounds formalize the idea that a set lies entirely to the “right” of some point. They are the dual notion to upper bounds and are used in defining infimum.

Examples:

• In $(\mathbb{R},\le)$, the set $S=(0,1)$ has lower bounds $\ell=0$, $\ell=-1$, and in fact every $\ell\le 0$.
• In $(\mathbb{R},\le)$, the set $S=\{x\in\mathbb{R}: x\ge 2\}$ has lower bounds $\ell=2$ and every $\ell\le 2$.
• In $(\mathbb{Z},\le)$, the set $S=\{n\in\mathbb{Z}: n\ge 0\}$ has lower bounds $\ell=0$, $\ell=-5$, etc.