So it's worth noting that the inductive hypothesis isn't false for all values of n - namely, it is true for n = 0,1.
But beyond that, in mathematical logic, for p -> q, you can still prove this to be true, even if p is false. In fact, if you can prove p is always false, p -> q is 'vacuously true' - it's true because you can never come up with an example of 'p and not q'.
So mathematically, there's no problem with assuming an induction hypothesis that's actually false here. The real problem occurs is that the result doesn't follow from the assumption because the induction step of the 'proof' sneakily assumes that n >= 3, i.e., the induction step simply doesn't work for the n = 2 case.
But beyond that, in mathematical logic, for p -> q, you can still prove this to be true, even if p is false. In fact, if you can prove p is always false, p -> q is 'vacuously true' - it's true because you can never come up with an example of 'p and not q'.
So mathematically, there's no problem with assuming an induction hypothesis that's actually false here. The real problem occurs is that the result doesn't follow from the assumption because the induction step of the 'proof' sneakily assumes that n >= 3, i.e., the induction step simply doesn't work for the n = 2 case.