1. Prove that for .
Proof. We use induction on . For , . Now suppose that for some , . So we want to show that . Now . Can we somehow use the fact that for all ? We could flip it so that for all . And so . Or another approach may be to show that ? The denominator of the second term is larger, indicating that it would probably be smaller.
2. Show that for , and deduce that .
For this one you would use the AM-GM inequality? Because the LHS is . So taking we can use the AM-GM inequality?