Definition:Infimum of Sequence

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This page is about infima of sequences which are bounded below. For other uses, see Definition:Infimum.

Definition

A special case of an infimum of a mapping is an infimum 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 an infimum.


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

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


Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the greatest lower bound of $T$ and denoted $\map {\operatorname {glb} } T$ or $\map {\operatorname {g.l.b.} } T$.

Some sources refer to the infimum of a set as the infimum on a set.


Also defined as

Some sources refer to the infimum as being the lower bound.

Using this convention, any element less than this is not considered to be a lower bound.


Linguistic Note

The plural of infimum is infima, although the (incorrect) form infimums can occasionally be found if you look hard enough.


Also see