# Definition:Supremum of Sequence

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

## Definition

A special case of a supremum of a mapping is a **supremum 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 a supremum.

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

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

## Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the **least upper bound of $T$** and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

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

Some sources refer to the **supremum of a set** as the **join of the set** and use the notation $\bigvee S$.

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

Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a **supremum** as an **upper limit**.

## Also defined as

Some sources refer to the supremum as being ** the upper bound**.

Using this convention, any element greater than this is not considered to be an upper bound.

## Linguistic Note

The plural of **supremum** is **suprema**, although the (incorrect) form **supremums** can occasionally be found if you look hard enough.