Let (an) and (bn) be bounded sequences, with bn̸ = 0 for all n. Consider thefour sequences(i) (an) (ii) (bn) (iii) (anbn) (iv) (an/bn)(a) Which of the sequences (i)–(iv) have convergent subsequences? Justifyyour answer. [6 marks](b) Suppose also that (an) and (bn) are monotonic. Which of the sequences(i)–(iv) must be convergent? Justify your answer.

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.)

A strictly monotonic increasing sequence is bounded below, then we can conclude that

Show from first principles that (an) is convergent, where an = cos(n) sin (n)/nfor all n ≥ 1. [3 marks](b) Let ℓ, m ∈ R and suppose that (cn) and (dn) are sequences such that(cn + dn) → ℓ + m. Show that (cn) → ℓ if and only if (dn) → m. Anyresults you use must be accurately stated. [4 marks](c) If (xn + yn) is a convergent sequence, must (xn − yn) also be convergent?Give a proof or a counterexample, with justification.

Let an = 6n3n + 1.(a)Determine whether {an} is convergent.convergentdivergent    (b)Determine whether ∞n = 1an is convergent.convergentdivergent

Problem 4. Prove that if ∑∞n=1 an converges absolutely and {bn}n≥1 is a bounded sequence,then the series ∑∞n=1 anbn also converges. (Hint: Use the Cauchy criterion.)