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.

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.

Define (xn)n≥1 by x1 := 0 and xn+1 := (xn + 3)/4 for n ≥ 1. Prove that the sequence (xn)n≥1 isconvergent and calculate the limit limn→∞ xn

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

For each n ∈ N, define fn : [0, 1] → R byfn(x) =(xn cos(1/x) if x > 0,0 if x = 0.11. Prove that fn ∈ R[0, 1] for all n ∈ N

Prove by mathematical induction that11×2×3+12×3×4+ ⋯ +1n(n+1)(n+2)=14−12(n+1)(n+2).