Math 104 homework 3 let sn and tn be cauchy sequences

Question

Math 104: Homework 3-

1. Let (a_n) be a sequence defined according to a₁ = t where t > 0, and a_n+1 = 2a_n/(1+a_n) for n ∈ N. Prove that a_n → 1 as n → ∞.

2. Let (s_n) and (t_n) be Cauchy sequences defined on R, and let (un) be a sequence defined as u_n = as_n + bt_n for all n, where a, b ∈ R. 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.

3. Let (s_n) be a sequence defined by s_n = (-1)ⁿ(1 + 1/n). Prove that lim sup s_n = 1 and lim infs_n = -1.