Bounded below

A bounded below set is a subset $A\subseteq\mathbb R$ for which there exists $m\in\mathbb R$ such that $m\le x$ for every $x\in A$; such an $m$ is called a lower bound of $A$.

This notion uses the order axioms and is the basic hypothesis needed to define the infimum . A set can be bounded below without having a minimum .

Examples:

• $A=(0,1)$ is bounded below (for example, by $m=0$).
• $A=\mathbb R$ is not bounded below.