Bounded below

Let $(X,\le)$ be an ordered set and let $S\subseteq X$. The set $S$ is bounded below if there exists an element $\ell\in X$ such that

Equivalently, $S$ is bounded below iff $S$ has a lower bound.

Boundedness below is the hypothesis needed to speak meaningfully about $\inf S$.

Examples:

• $(0,1)\subseteq\mathbb{R}$ is bounded below (e.g. by $\ell=0$).
• $\{x\in\mathbb{R}: x\ge -7\}$ is bounded below (e.g. by $\ell=-7$).
• The set $\{ -n : n\in\mathbb{N}\}\subseteq\mathbb{R}$ is not bounded below.