5. (Squeeze Theorem for Sequences). Assume that lim an = lim on = L 1-+00 7-+00 and that an

5. (Squeeze Theorem for Sequences). Assume that lim an = and that an bn cn for all n e N. Prove that: lim bn = L limcn=L 6. Prove that if lim an = O and that (bn) is a bounded sequence, then lim a,zbn = O 7. Show that the sequence (n)X1 diverges. 8. Finish the proof of Theorem 5.6 (ii). That is, show that if (an) is an unbounded decreasing sequence then: lim an = OC 9. (Bonus Question) Prove that a bounded increasing sequence is a Cauchy sequence. Conclude that Cauchy's Convergence Criterion implies the Monotone Convergence Theorem.