# Definition:Infimum of Set/Real Numbers

*This page is about Infimum of Subset of Real Numbers. For other uses, see Infimum.*

## Definition

Let $T \subseteq \R$.

A real number $c \in \R$ is the **infimum of $T$ in $\R$** if and only if:

- $(1): \quad c$ is a lower bound of $T$ in $\R$
- $(2): \quad d \le c$ for all lower bounds $d$ of $T$ in $\R$.

If there exists an **infimum** of $T$ (in $\R$), we say that **$T$ admits an infimum (in $\R$)**.

The **infimum** of $T$ is denoted $\inf T$ or $\map \inf T$.

## Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the **greatest lower bound of $T$** and denoted $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.

Some sources refer to the **infimum of a set** as the **infimum on a set**.

Some sources introduce the notation $\displaystyle \inf_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to a **infimum** as a **lower limit**.

## Also defined as

Some sources refer to the infimum as being ** the lower bound**.

Using this convention, any element less than this is not considered to be a lower bound.

## Examples

### Example 1

The subset $S$ of the real numbers $\R$ defined as:

- $S = \set {1, 2, 3}$

admits an infimum:

- $\inf S = 1$

### Example 2

The subset $T$ of the real numbers $\R$ defined as:

- $T = \set {x \in \R: 1 \le x \le 2}$

admits an infimum:

- $\inf T = 1$

### Example 3

The subset $V$ of the real numbers $\R$ defined as:

- $V := \set {x \in \R: x > 0}$

admits an infimum:

- $\inf V = 0$

## Also see

- Characterizing Property of Infimum of Subset of Real Numbers
- Definition:Supremum of Subset of Real Numbers
- Supremum and Infimum are Unique

## Linguistic Note

The plural of **infimum** is **infima**, although the (incorrect) form **infimums** can occasionally be found if you look hard enough.

## Sources

- 1919: Horace Lamb:
*An Elementary Course of Infinitesimal Calculus*(3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence - 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.34$. Definition - 1970: Arne Broman:
*Introduction to Partial Differential Equations*... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.5$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.6$: Supremum and Infimum - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**order properties**(of real numbers) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**bound**