# Continuum Property/Proof 2

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## 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.

$\blacksquare$