# Definition:Supremum of Real Sequence

## Definition

Let $\sequence {x_n}$ be a real sequence.

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.