# Definition:Infimum of Real Sequence

*This page is about infima of real sequences which are bounded below. For other uses, see Definition:Infimum.*

## Definition

Let $\left \langle {x_n} \right \rangle$ be a real sequence.

Let $\left\{{x_n: n \in \N}\right\}$ admit an infimum.

Then the **infimum** of $\left \langle {x_n} \right \rangle$) is defined as:

- $\displaystyle \inf \left({\left \langle {x_n} \right \rangle}\right) = \inf \left({\left\{{x_n: n \in \N}\right\}}\right)$

## Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the **greatest lower bound of $T$** and denoted $\map {\operatorname {glb} } T$ or $\map {\operatorname {g.l.b.} } T$.

Some sources refer to the **infimum of a set** as the **infimum on a set**.

## Also defined as

Some sources refer to the infimum as being ** the lower bound**.

Using this convention, any element less 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.