Least Upper Bound Property

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 $\R$, or the completeness axiom.

Proof
A direct consequence of Dedekind's Theorem.

Also known as
The continuum property of $\R$ is also known as the completeness axiom.

Also see
Not to be confused with:
 * The Continuum Hypothesis
 * The Continuity Property