# Continuum Property

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## Theorem

The **continuum property (of the set of real numbers $\R$)** is a complementary pair of theorems whose subject is the real number line:

### Least Upper Bound Property

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above.

Then $S$ admits a supremum in $\R$.

This is known as the **least upper bound property** of the real numbers.

### Greatest Lower Bound Property

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below.

Then $S$ admits an infimum in $\R$.

This is known as the **greatest lower bound property** of the real numbers.

## Also presented as

The **Continuum Property** can also be stated as:

- The set $\R$ of real numbers is Dedekind complete.

## Also known as

The **Continuum Property of $\R$** is also known as:

- the
**completeness axiom** - the
**completeness property** - the
**completeness postulate**

## Also see

Not to be confused with:

## Sources

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function: Theorem $\text{A}$ - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.4$: The Continuum Property