Continuum_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. 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 $\R$'''. This can also be stated as The set $\R$ of real numbers is Dedekind complete.