Dedekind theorem

on the continuity of the real axis

For any cut $A|B$ of the set of real numbers (see Dedekind cut) there exists a real number $\alpha$ which is either the largest in the class $A$ or the smallest in the class $B$. This statement is also known as the Dedekind principle (axiom) of continuity of the real axis (cf. Real number). The number $\alpha$ is the least upper bound of $A$ and the greatest lower bound of $B$.