Prove that the rational numbers are dense in the reals. I.e., if with , then there exists an such that . It follows that there are then infinitely many such.
Proof. Since , we know . Therefore, there exists an such that
We also know (I.3.12, Exercise #4) that there exists such that . Putting these together we have,
Since we have . Hence, letting we have found such that
This then guarantees infinitely many such rationals since we can just replace by (and note that ) and apply the theorem again to find such that . Repeating this process we obtain infinitely many such rationals