Definition:Infimum of Set/Real Numbers

Definition
Let $T \subseteq \R$.

A real number $c \in \R$ is the infimum of $T$ in $\R$ iff:


 * $(1): \quad c$ is a lower bound of $T$ in $S$
 * $(2): \quad d \le c$ for all lower bounds $d$ of $T$ in $S$.

The infimum of $T$ is denoted $\inf T$.

If there exists an infimum of $T$ (in $\R$), we say that $T$ admits an infimum (in $S$).

Also known as
The infimum of $T$ is often called the greatest lower bound of $T$ and denoted $\operatorname{glb} \left({T}\right)$ or $\operatorname{g.l.b.} \left({T}\right)$.

Some sources refer to the infimum as being the lower bound. Using this convention, any number smaller than this is not considered to be a lower bound.

Linguistic Note
The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.

Also see

 * Definition:Supremum of Subset of Real Numbers


 * Supremum and Infimum are Unique