Definition:Supremum of Sequence

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This page is about suprema of sequences which are bounded above. For other uses, see Definition: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 $\left({T, \preceq}\right)$ be an ordered set.

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


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.


Also see