All rational numbers have the property that their square is not 2 == 2 not in {r^2 for r in rationals} == sqrt(2) not in {r for r in rationals} ==

All rational numbers have the property that their square is not 2 == 2 not in {r^2 for r in rationals} == sqrt(2) not in {r for r in rationals} == sqrt(2) not in rationals It seems to me that you handwaved away the entire proof in your first sentence, without really proving anything.