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 $\bigvee T$ or $\ds \bigvee_{y \mathop \in T} y$.
Some sources introduce the notation $\ds \sup_{y \mathop \in T} y$, 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.