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 producer of the $\tuple{L,R}$.

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 see

 * Axiom of Continuity