1.25 Remark. Ifx > 1 and x ~ N, then there is ann E N such thatn < x < n+l.

PROOF. By the Archimedean Principle, the set E = {m EN: x < m} is nonempty. Hence by the Well-Ordering Principle, E has a least element, say mo.

Set n = rna-I. Since mo E E, n+l = mo > x. Since mo is least, n = mo-l :::; x.

Since x ~ N, we also have n =I- x. Therefore, n < x < n + 1. I Using this last result, we can prove that the set of irrationals is nonempty.