Dedekind's Theorem

Theorem
Let $\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\R$.

Then there exists a unique real number which is a producer of $\tuple {L, R}$.

Thus it is proved that the totally ordered set $\R$ is Dedekind complete, and that is why it is referred to as the continuum.

Also known as
Dedekind's Theorem is also known as the completeness theorem for the real numbers.

Also see

 * Axiom:Axiom of Continuity


 * Continuum Property