5.2 Prove by induction on that for .
Proof: We use induction on . Base case: For , and and so . Inductive step: Suppose now as inductive hypothesis that for some . Then (by induction hypothesis). So we will have proved that if we can prove that . But , and since , so that . Hence and so we have deduced that as required to complete the inductive step. Conclusion: Hence, by induction, for all .
Yes, you are correct. It's just that as written, your proof is kind of hard to read. Try formatting it better so that the reader does not get lost in the sea of words and inequalities. I believe that's why TPH's first post was like that. He probably read through your post once and realized that he had to read it again to really get what you were saying--but thought it easier to just tell you what to do than to read through it again. Your second comment kind of forced him to read through it carefully to see that you did, in fact, do what he suggested in the first place.