I'm trying to work this out in my head, but I believe that this holds if m is a rational such that its numerator and denominator are not both perfect squares (e.g., √(2/3) is irrational but √(4/9) is rational). Am I wrong?
When this proof has been presented to me in the past, I believe we assume that m is a positive integer. This is mentioned in one of the comments, but I agree that it should be more clear in statement of the theorem.
Good point! The only part we need to check is the base case. Since the result holds for n=0, I'd say you're right.