# Continuum Property/Proof 2

## Theorem

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.

Similarly, 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 sometimes called the greatest lower bound property of the real numbers.

The two properties taken together are called the continuum property of $\R$.

## Proof

A direct consequence of Dedekind's Theorem.

$\blacksquare$