In the following argument to show that √2 is irrational, give proofs appropriate for steps 3, 5 and 6.
1. Assume that √2 is rational.
2. Define the set S to be the set of positive integers with the following property:
n is in S if and only if n√2 is an integer.
3. Show that the set S contains at least one positive integer.
4. Define the integer k to be the smallest positive integer in S
5. Show that (√2-1)k is in S.
6. Show that steps 4 and 5 are contradictory and hence that √2 is irrational.
Prove that is rational if and only if is rational.
Use an argument similar to that of part (i) to prove that and are irrational.