# Definition:Supremum of Set/Real Numbers

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

## Definition

Let $T \subseteq \R$ be a subset of the real numbers.

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

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

If there exists a **supremum** of $T$ (in $\R$), we say that:

**$T$ admits a supremum (in $\R$)**or**$T$ has a supremum (in $\R$)**.

The **supremum** of $T$ is denoted $\sup T$ or $\map \sup T$.

### Definition by Propositional Function

Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\map R j$ be a propositional function of $j \in I$.

Then we can define the **supremum of $\family {a_j}_{j \mathop \in I}$** as:

- $\displaystyle \sup_{\map R j} a_j := \text{ the supremum of all $a_j$ such that $\map R j$ holds}$

If more than one propositional function is written under the supremum sign, they must *all* hold.

## Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the **least upper bound of $T$** and denoted $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

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

Some sources refer to the **supremum of a set** as the **join of the set** and use the notation $\bigvee S$.

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

Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a **supremum** as an **upper limit**.

## Also defined as

Some sources refer to the supremum as being ** the upper bound**.

Using this convention, any element greater than this is not considered to be an upper bound.

## Examples

### Example 1

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

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

admits a supremum:

- $\sup S = 3$

### Example 2

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

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

admits a supremum:

- $\sup T = 2$

### Example 3

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

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

does not admit a supremum.

### Example 4

Consider the set $A$ defined as:

- $A = \set {3, 4}$

Then the supremum of $A$ is $4$.

However, $A$ contains no element $x$ such that:

- $3 < x < 4$.

## Also see

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

- Results about
**suprema**can be found here.

## Linguistic Note

The plural of **supremum** is **suprema**, although the (incorrect) form **supremums** 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): $\S 2.5$: 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: Definition $1.1.2$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.6$: Supremum and Infimum - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**bound**