Upper Bound¶

Definition¶

Let $E$ be a subset of $\mathbb{R}\text{,}$ and let $M$ be a real number. We say that $M$ is an upper bound for $E$ if we have $x \leq M$ for every element $x$ in $E\text{.}$

Interpretation¶

An upper bound for a set $E$ is a number $M$ that no element of $E$ exceeds. It controls the set from above without having to belong to it and without having to be close to it.

A number line with a set E made of the interval from -2 to 0.5 and the point 1.1, with its upper bounds shaded from 1.1 onwards; the least upper bound 1.1 is the largest element of E.

Here $E$ consists of the interval $[-2,0.5]$ and the isolated point $1.1$ . Every $M\ge1.1$ is an upper bound, including the marked $M_1$ , $M_2$ and $M_3$ , so a set with one upper bound has infinitely many. The least of them, $\sup E=1.1$ $1.1$ , is also $\max E$ , because $1.1$ belongs to $E$ .

The inequality $x\le M$ must hold for every $x\in E$ with the same $M$ . To show that $M$ is not an upper bound, one element is enough: a single $x\in E$ with $x>M$ $M$ . Equality is allowed, so $1$ is an upper bound for $(0,1)$ as well as for $[0,1]$ . For the empty set the condition holds vacuously, and every real number is an upper bound.

A set with an upper bound is called bounded above; a set with both an upper and a lower bound is bounded. Applied to the terms of a sequence, the same idea gives a bounded sequence. The least upper bound is the supremum, which exists for every nonempty set that is bounded above.

A common slip is to confuse an upper bound with a maximum. An upper bound need not lie in the set and is never unique; the maximum lies in the set and is unique when it exists.

Key Results¶

If $M$ is an upper bound for $E$ , then so is every $M'\ge M\text{;}$ hence upper bounds are never unique.
Upper bounds need not belong to the set.

Examples¶

The set $(0,1)$ has upper bound $1\text{.}$
Any number greater than $1$ is also an upper bound for $(0,1)\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 3

Aliases: majorant Keywords: bounded above, upper bounds Symbols: element of, real numbers

Last updated: 20 October 2026 03:02