Infimum¶

Definition¶

Let $E\subseteq\mathbb{R}$ be nonempty and bounded below. The infimum of $E$ is its greatest lower bound, denoted by $\inf E\text{.}$

Interpretation¶

The infimum is the greatest value that does not exceed any element of a set: among all lower bounds of $E$, it is the largest. It provides a precise way to describe how far down a set reaches, even when no minimum exists.

For $E=\{1,\tfrac{1}{2},\tfrac{1}{3},\ldots\}$$\{1,\frac{1}{2},\frac{1}{3},\dots \}$, every number $\le0$ is a lower bound, and no positive number $c$ is one, since $\tfrac{1}{n}<c$ once $n>\tfrac{1}{c}$$\frac{1}{c}$. So the shaded lower bounds end exactly at $0$, and $\inf E=0$ although $0\notin E$$E$; the set has no minimum.

The definition has two parts, and the interactive figure tests both. A number $m$ is the infimum of $E$ if and only if $m$ is a lower bound of $E$ and, for every $\varepsilon>0$$0$, some $x\in E$ satisfies $x<m+\varepsilon$$\epsilon$. The first part says that $m$ is low enough; the second says that nothing larger works, because $m+\varepsilon$ is not a lower bound. This is the usual way to verify an infimum, as for $(0,1)$ in the Examples.

A common slip is to think that the infimum must belong to the set. It does exactly when the set has a minimum, and then $\inf E=\min E$$\min E$. Another is to expect an infimum in every number system: in $\mathbb{Q}$, the set $\{x\in\mathbb{Q}\colon x>0,\ x^2>2\}$ is bounded below but has no greatest rational lower bound, since its infimum in $\mathbb{R}$ is $\sqrt{2}$. The infimum mirrors the supremum: $\inf E=-\sup(-E)$$-\sup (-E)$, where $-E=\{-x\colon x\in E\}$$E\}$.

Key Results¶

• By the greatest-lower-bound property (see real number) of $\mathbb{R}\text{,}$ every nonempty subset of $\mathbb{R}$ that is bounded below has a unique infimum.
• An infimum need not belong to the set; it is a minimum exactly when it is attained.
• In the extended real numbers, one uses the conventions $\inf E=-\infty$ for a nonempty set with no lower bound and $\inf\varnothing=+\infty\text{;}$ neither $+\infty$ nor $-\infty$ is a real number.

Examples¶

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

Prerequisites¶

• Lower Bound

Related Concepts¶

• Supremum
• Minimum of a Set
• Bounded Set
• Bounded Function

Sources¶

• Math I: Calculus · Lecture 3

Aliases: greatest lower bound, inf Symbols: element of, epsilon, infinity, rational numbers, real numbers, square root, subset

Last updated: 20 October 2026 03:02