If (xn) is a bounded sequence in R then the following statements about a numberx∗ ∈ R are equivalent

Let (xn) be a bounded sequence in R. Show that there exist subsequences (xnk ) and(xmk ) of (xn) such thatlimk→∞ xnk = lim sup xn and limk→∞ xmk = lim inf xn.

The sequence x_n={1, 2, 3, 4, 5, 5, 6, ...} is .......... and ............. sequence.1 pointan increasing and unboundeda non-decreasing and unboundeda decreasing and unboundedan increasing and bounded

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n. [5 marks](b) Suppose that a sequence (cn) satisfies |cn| > C|cn−1| for all n ≥ 1, whereC > 1. Show that (|cn|) → ∞. Does it follow that either (cn) → ∞ or(cn) → −∞? Justify your answer. [6 marks](c) If |dn| > |dn−1| for all n, does it follow that (|dn|) → ∞? Justify youranswer with a proof or a counterexample.

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.

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.)