# Definition:Bounded Above Set/Real Numbers

*This page is about subsets of the real numbers which are bounded above. For other uses, see Definition:Bounded Above.*

## Contents

## Definition

Let $\R$ be the set of real numbers.

A subset $T \subseteq \R$ is **bounded above (in $\R$)** if and only if $T$ admits an upper bound (in $\R$).

### Unbounded Above

$T \subseteq \R$ is **unbounded above (in $\R$)** if and only if it is not bounded above.

## Examples

### Example: $\hointl \gets 2$

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

- $I = \hointl \gets 2$

is bounded above by, for example, $2$, $3$ and $4$, of which the supremum is $2$.

$2$ is also the greatest element of $I$.

The set of all upper bounds of $I$ is:

- $\closedint 2 \to$

### Example: $\openint 0 1$

Let $I$ be the open real interval defined as:

- $I := \openint 0 1$

Then $I$ is bounded above by, for example, $1$, $2$ and $3$, of which $1$ is the supremum.

However, $I$ does not have a greatest element.

### Example: $\set {-1, 0, 2, 5}$

Let $I$ be the set defined as:

- $I := \set {-1, 0, 2, 5}$

Then $I$ is bounded above by, for example, $5$, $6$ and $7$, of which the supremum is $5$.

$5$ is also the greatest element of $I$.

### Example: $\openint 3 \to$

Let $I$ be the unbounded open real interval defined as:

- $I := \openint 3 \to$

Then $I$ is not bounded above.

Hence $I$ does not admit a supremum, and so does not have a greatest element.

### Example: $\closedint 0 1$

Let $I$ be the closed real interval defined as:

- $I := \closedint 0 1$

Then $I$ is bounded above by, for example, $1$, $2$ and $3$, of which $1$ is the supremum.

$I$ is also the greatest element of $I$.

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.1$: Real Numbers - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.2$: The Continuum Property - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 10$: The well-ordering principle - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**bound**