Infimum

definitionreal-analysis

Definition

We say that $m\in \mathbb{R}$ is the infimum of a set $S\subset \mathbb{R}$ if the following are true,

1. $m$ is a lower bound of $S$ (therefore $S$ is bounded below).
2. For any lower bound $m^{\prime }$ of $S$, $m^{\prime }\le m$ ($m$ is unique).

$m$ is also called the greatest lower bound.

We denote the infimum of a set $S$, $\inf (S)$.

Examples