# Definition:Infimum of Sequence

*This page is about Infimum of Sequence. For other uses, see Infimum.*

## Definition

A special case of an infimum of a mapping is an **infimum of a sequence**, where the domain of the mapping is $\N$.

Let $\struct {T, \preceq}$ be an ordered set.

Let $\sequence {x_n}$ be a sequence in $T$.

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

Then the **infimum** of $\sequence {x_n}$) is defined as:

- $\map \inf {\sequence {x_n} } = \map \inf {\set {x_n: n \in \N} }$

## 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 $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.

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

Some sources introduce the notation $\ds \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to an **infimum** as a **lower limit**.

## 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.