Dedekind's Theorem

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

Then:
 * $\exists \alpha \in \R: \forall x \in \R: \paren {x < \alpha \implies x \in L} \text { and } \paren {x > \alpha \implies x \in R}$

such that:
 * $\alpha$ may be an element of either $L$ or $R$

and
 * there exists exactly one such $\alpha$.

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