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Homesupremum

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# supremum

The *supremum* of a set $X$ having a partial order is the least upper bound of $X$ (if it exists) and is denoted $\sup{X}$.

Let $A$ be a set with a partial order $\leqslant$, and let $X\subseteq A$. Then $s=\sup X$ if and only if:

- 1.
For all $x\in X$, we have $x\leqslant s$ (i.e. $s$ is an upper bound).

- 2.
If $s^{{\prime}}$ meets condition 1, then $s\leqslant s^{{\prime}}$ ($s$ is the

*least*upper bound).

There is another useful definition which works if $A=\mathbb{R}$ with $\leqslant$ the usual order on $\mathbb{R}$, supposing that s is an upper bound:

$s=\sup X\text{ if and only if }\forall\varepsilon>0,\exists x\in X:s-% \varepsilon<x.$ |

Note that it is not necessarily the case that $\sup X\in X$. Suppose $X={]0,1[}$, then $\sup X=1$, but $1\not\in X$.

Note also that a set may not have an upper bound at all.

Keywords:

real analysis

Related:

Infimum, MinimalAndMaximalNumber, InfimumAndSupremumForRealNumbers, ExistenceOfSquareRootsOfNonNegativeRealNumbers, LinearContinuum, NondecreasingSequenceWithUpperBound, EssentialSupremum

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

06A06*no label found*

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