WLOG assume . By the Axiom of Archimedes there exists with . Consider the set of natural numbers with . It is non-empty, since some natural number exceeds , and so it has a least element : this is where the Well-Ordering Principle enters. Then , while if we would get , contradicting minimality. Hence the rational lies strictly between and .

The irrationals are dense as well: choosing a positive rational with , the number is irrational and satisfies .