Definition:Infimum of Sequence

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This page is about Infimum of Sequence. For other uses, see 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 $\struct {T, \preceq}$ be an ordered set.

Let $\sequence {x_n}$ be a sequence in $T$.

Let $\set {x_n: n \in \N}$ admit an infimum.

Then the infimum of $\sequence {x_n}$) is defined as:

$\map \inf {\sequence {x_n} } = \map \inf {\set {x_n: n \in \N} }$

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 $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.

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

Some sources introduce the notation $\ds \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to an infimum as a lower limit.

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.