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# infimum

The *infimum* of a set $S$ is the greatest lower bound of $S$ and is denoted $\inf(S)$.

Let $A$ be a set with a partial order $\leq$, and let $S\subseteq A$. For any $x\in A$, $x$ is a lower bound of $S$ if $x\leq y$ for any $y\in S$. The infimum of $S$, denoted $\inf(S)$, is the greatest such lower bound; that is, if $b$ is a lower bound of $S$, then $b\leq\inf(S)$.

Note that it is not necessarily the case that $\inf(S)\in S$. Suppose $S=(0,1)$; then $\inf(S)=0$, but $0\not\in S$.

Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.

Keywords:

real analysis

Related:

Supremum, LebesgueOuterMeasure, MinimalAndMaximalNumber, InfimumAndSupremumForRealNumbers, NondecreasingSequenceWithUpperBound

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

06A06*no label found*03D20

*no label found*

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