Infimum - Knowlpedia

An infimum of a nonempty set $A\subseteq\mathbb R$ that is bounded below is a real number $t\in\mathbb R$ such that:

$t$ is a lower bound of $A$ (i.e., $t\le x$ for all $x\in A$ ), and
for every lower bound $\ell$ of $A$ , one has $\ell\le t$ .

Existence and comparison

The infimum is the “greatest lower bound” and may exist even when $A$ has no minimum. Using the completeness axiom, every nonempty bounded-below set of real numbers has an infimum.

Examples

If $A=(0,1)$ , then $\inf A=0$ .
If $A=\{\tfrac1n : n\in\mathbb N,\ n\ge 1\}$ , then $\inf A=0$ (even though $0\notin A$ ).