Math 104: Homework 3 (due February 11)

1. Ross exercise 8.10

2. Ross exercise 9.6

3. Let (a_{n}) be a sequence defined according to a_{1} = t where t > 0, and a_{n+1} = 2a_{n}/(1 + a_{n}) for n ∈ . Prove that a_{n} → 1 as n → ∞.

4. Let (s_{n}) and (t_{n}) be Cauchy sequences defined on , and let (u_{n}) be a sequence defined as u_{n} = as_{n} + bt_{n} for all n, where a, b ∈ . By using the definition of a Cauchy sequence only, without assuming that limits of (s_{n}) and (t_{n}) exist, prove that (u_{n}) is a Cauchy sequence.

5. Ross exercise 9.10

6. Ross exercise 10.6

7. Ross exercise 10.8

8. Let (s_{n}) be a sequence defined by s_{n} = (−1)^{n}(1 + \frac{1}{n}). Prove that lim sup s_{n} = 1 and lim inf s_{n} = −1.