It sounds like you don't understand how an inductive proof works. Steps 7 and 8 are perfectly fine. Everything up to that point is perfectly fine. The problem

It sounds like you don't understand how an inductive proof works. Steps 7 and 8 are perfectly fine. Everything up to that point is perfectly fine. The problem is only in step 9, exactly in that "with the exception of k < 3" (actually k<2) - because the trivially true base case of k=1 is that exception. The induction step is correct, it just doesn't work for the only available true base case.