# Definition:Supremum of Real Sequence

*This page is about suprema of real sequences which are bounded above. For other uses, see Definition:Supremum.*

## Definition

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

Let $\left\{{x_n: n \in \N}\right\}$ admit a supremum.

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

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

## 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 $\map {\operatorname {lub} } T$ or $\map {\operatorname {l.u.b.} } T$.

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

## 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.