Lower Bound and Greatest Lower Bound

Let $S\subseteq\mathbb{R}$. A number $\ell\in\mathbb{R}$ is called a lower bound of $S$ if $\ell\le s$ for every $s\in S$. If $S$ is nonempty and bounded below, then a number $m\in\mathbb{R}$ is called the greatest lower bound, or infimum, of $S$ if:

$$\text{(i) } m \text{ is a lower bound of } S, \qquad \text{and} \qquad \text{(ii) every lower bound } \ell \text{ of } S \text{ satisfies } \ell\le m.$$

In that case one writes $m=\inf S$.