Supremum¶

Definition¶

Let $E\subseteq\mathbb{R}$ be nonempty and bounded above. The supremum of $E$ is its least upper bound, denoted by $\sup E\text{.}$

Interpretation¶

The supremum of a set is its least upper bound: the smallest number that no element of the set exceeds. It describes where a set ends on the right even when no element sits there, that is, even when the set has no maximum.

The open interval from 0 to 1 on a number line, with its upper bounds shaded from 1 onwards and an arrow marking 1 as the supremum, which is not in the set.

For $E=(0,1)$ $(0, 1)$ , every number from $1$ onwards is an upper bound, such as the marked $1.6$ and $2.2$ ; together they form the shaded ray $[1,\infty)$ . Its left end is the least upper bound, so $\sup E=1$ $1$ , although $1\notin E$ $E$ .

Move the candidate $u$ : every number at or above $\sup E$ is an upper bound, and for every $\varepsilon>0$ some element of $E$ lies within $\varepsilon$ below it, which is what makes it the least one.

Being the least upper bound has two parts: $u=\sup E$ is an upper bound, and no smaller number is one. The second part is usually checked with an $\varepsilon$ : for every $\varepsilon>0$ some $x\in E$ satisfies $x>u-\varepsilon$ $ε$ . For $(0,1)$ and $0<\varepsilon<1$ $1$ , the element $1-\tfrac{\varepsilon}{2}$ lies above $1-\varepsilon$ $ε$ .

The existence of suprema is the completeness of the real numbers, and several results depend on it: the monotone convergence theorem identifies its limit as a supremum, and the $n$ -th root is defined as one. The mirror notion is the infimum.

A common slip is to treat $\sup E$ as the largest element of $E$ . It is the largest element exactly when it belongs to $E$ , and then $\sup E=\max E$ $max E$ ; for $(0,1)$ it does not belong to the set.

Key Results¶

A supremum need not belong to the set.
By the least-upper-bound property (see real number) of $\mathbb{R}\text{,}$ every nonempty subset of $\mathbb{R}$ that is bounded above has a unique supremum.
In the extended real numbers, one uses the conventions $\sup E=+\infty$ for a nonempty set with no upper bound and $\sup\varnothing=-\infty\text{;}$ neither $+\infty$ nor $-\infty$ is a real number.

Examples¶

The supremum of $(0,1)$ is $1\text{.}$
The supremum of $\{x\in\mathbb{R}\colon x<2\}$ is $2\text{.}$

Prerequisites¶

Upper Bound

Sources¶

Math I: Calculus · Lecture 3

Aliases: least upper bound, sup Symbols: element of, epsilon, infinity, real numbers, subset

Last updated: 20 October 2026 03:02