Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn. (0.1)NOTE) Let xn := (−1)n and yn := (−1)n+1 for n ≥ 1. Thenlim infn→∞ (xn + yn) = 0 > −2 = lim infn→∞ xn + lim infn→∞ yn.Hence, the equality in (0.1) does not hold in general.(3-4) Recall that the Fibonacci sequence (zn)n≥1 is defined byz1 = z2 = 1 and zn+2 = zn+1 + zn for all n ≥ 1.Define a sequence (xn)n≥1 byxn = znzn+1

Let (xn)n≥1 and (yn)n≥1 be bounded sequence. Prove thatlim infn→∞ (xn + yn) ≥ lim infn→∞ xn + lim infn→∞ yn

Problem 2. Let {an}n≥1 be a sequence. Prove that if {akn }n≥1 is a subsequence of {an}n≥1that converges to L ∈ R, thenlim infn→∞ an ≤ L ≤ lim supn→∞an.(Hint: As {an}n≥1 is not necessarily bounded, lim infn→∞ an and lim supn→∞an might be equal to±∞ here. However, in these cases, the proof is much easier.)

Calculate the following quantities:(a) lim infn→∞ (−1)n3 + 1n

Prove that the sequence (xn)n≥1 converges and compute limn→∞ xn

Define (xn)n≥1 by x1 := 1 and xn+1 := (3xn + 3)/4 for n ≥ 1. Prove thatlimn→∞ xn = 3.(Hint: First show that xn ≤ 3 for all n ≥ 1.)