In mathematics, a

**supremum**is a largest possible quantity (subject to given conditions), if such a thing exists; or otherwise, within a larger set of quantities, it is a minimal larger choice (if such exists). For example, under different tax systems there might be a 'largest' percentage

*tax rate that anyone would have to pay*; and this could mean that someone actually paid at that rate, or it might refer instead to the top rate that limits the percentage a very high earner might pay.

Table of contents |

2 Supremum of a poset 3 Comparison with Maximum 4 Least upper bound property |

## Supremum of a set of real numbers

In analysis the **supremum** or **least upper bound** of a set *S* of real numbers is denoted by sup(*S*) and is defined to be the smallest real number that is greater than or equal to every number in *S*.
An important property of the real numbers is that every nonempty set of real numbers that is bounded above has a supremum. This is sometimes called the *supremum axiom* and expresses the completeness of the real numbers.
If in addition we define sup(*S*) = -∞ when *S* is empty, and sup(*S*) = +∞ when *S* is not bounded above then *every* set of real numbers has a supremum (see extended real number line).

Examples:

- sup {
*x*in**R**: 0 < x < 1 } = 1 - sup {
*x*in**R**:*x*^{2}< 2 } = √2 - sup { (-1)
^{n}- 1/*n*:*n*= 1, 2, 3, ...} = 1

*S*does not have to belong to

*S*(like in these examples). If the supremum value belongs to the set then we can say there is a largest element in the set.

In general, in order to show that sup(*S*) ≤ *A*, one only has to show that *x* ≤ *A* for all *x* in *S*. Showing that sup(*S*) ≥ *A* is a bit harder: for any ε > 0, you have to exhibit an element *x* in *S* with *x* ≥ *A* - ε.

In functional analysis, one often considers the **supremum norm** of a bounded function *f* : *X* `->` **R** (or **C**); it is defined as

- ||
*f*||_{∞}= sup { |*f*(*x*)| :*x*∈*X*}

*See also*: infimum or *greatest lower bound*, limit superior.

## Supremum of a poset

It can easily be shown that, if*S*has a supremum, then the supremum is unique: if

*u*

_{1}and

*u*

_{2}are both suprema of

*S*then it follows that

*u*

_{1}<=

*u*

_{2}and

*u*

_{2}<=

*u*

_{1}, and since <= is antisymmetric it follows that

*u*

_{1}=

*u*

_{2}.

In an arbitrary partially ordered set, there may exist subsets which don't have a supremum.
In a lattice every nonempty *finite* subset has a supremum, and in a complete lattice every subset has a supremum.