1. (3 marks) Let p be a positive integer. You may assume without proof that nP _ n for all n E N. Use the precise definition of a limit diverging to too to prove that lim, con = to. 2. Suppose k and & are positive integers, and ak, ak-1, ..., do, be, be-1, ..., bo E R with ak # 0 and be # 0. Define (Sn) by an* + ak_ Ink-1+ ... + ain + do Sn bene + be- 1ne - 1 + ... + bin + bo (we assume that the denominator is not equal to zero for any n E N). Use the various limit laws from section 9 to prove the following results. Please indicate which theorems you are using (a) (4 marks) Prove that if k = 4, then lim sn = ak bk (b) (4 marks) Prove that if k l and * > 0, then lim sn = too. Be careful: Theorem 9.10 cannot be used here as some terms may be negative. 3. (4 marks) Exercise 9.10 (c). Use the precise definition of a limit in your proof, and not previously seen theorems. 4. (4 marks) Prove that the sequence 2 Sn = 1 - 5 - is increasing. 9.10 (c) Show that if lim sn = too and k