# Continuum Property

## 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: