Am I assuming too much here?
I don't understand why you're setting in your proof if is irrational.
To prove these, you just use the fact that is a field, i.e, closed under addition, subtraction, multiplication, and division.
If , then by closure, . Contradiction.
If , then by closure, . Contradiction.
He did NOT assume . He showed, first, that if you assume a/t is a rational number, then you arrive at t= r/s, a ration number and so a contradiction.
He then showed that if you assume at is rational, [b]then[b] you arrive at t= r/s, again a contradiction. His proof is perfectly valid.
To prove these, you just use the fact that is a field, i.e, closed under addition, subtraction, multiplication, and division.
If , then by closure, . Contradiction.
If , then by closure, . Contradiction.