Math 104 homework 3 let an be a sequence defined according

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 for n ∈ N as

Calculate the monotonic sequences

u_N = inf{s_n : n > N},           v_N = sup{s_n : n > N}

for each N ∈ N and hence determine lim inf s_n and lim sup s_n.