MATH 521, Spring 2014, Assignment 5Due date: Wednesday, April 2Problems for submission:1. Use the definition of the limit to prove that the sequences {an} withthe given terms converge to the indicated limits:(a) an=n22n2+ n − 1, L = 1/2(b) an=pn2+ n − n, L = 1/22. Consider a sequence {an} ⊂ R which is recursively defined byan+1= f(an). (1)(a) Prove that if there is some L ∈ R and a 0 ≤ c < 1 such thatan+1− Lan− L≤ cfor all n ∈ N, then limn→∞an= L.(b) Consider the sequence (1) with f(an) =1 − 2an3. Use the resultof part (a) to show that {an} converges to the limit L = 1/5regardless of the value of a1. What is the value of c?(c) Consider the sequence (1) with f(an) = 1 +11 + anand a1= 1.Use the result of part (a) to show that {an} converges to thelimit L =√2. What is the value of c? (Hint: first show that1 ≤ an≤ 2 implies 1 ≤ an+1≤ 2, or some comparable bound.)3. Consider the sequence {an} where an+2= an+1+ an, n ∈ N.(a) Consider the initial values a1= 1 and a2= 1. Explicitly de-termine up to the term a6and then prove that the sequencediverges. (Hint: Show it is monotonically increasing but has noupper bound.)(b) Consider the initial values a1= 1 and a2=1−√52. Explicitlydetermine up to the term a6. Then prove inductively thatan+1= 1 −√52!anand use this to prove that the sequence converges to the limitL = 0.4. Chapter #3, Question #20 in Rudin5. Determine the limit inferior and limit superior of the following se-quences (you do not need to rigorously prove the limits, but the limitvalues need to be exact):(a) an= (−1)nn + 1n − 1arctan(n), (b) an= sin(2n − 1)π2+1n(c) a2n= −12a2n−1, a2n+1= a2n+ 1, a1= 0.6. Consider a sequence of real numbers {an} which is bounded. Definethe set S to be the set of all subsequential limits of {an}.(a) Prove that s ∈ S if and only if, for every  > 0 and every N ∈ Nthere is an n > N so that |an− s| < .(b) Define S∗= sup(S). Prove that α ∈ R satisfies the following twoconditions if and only if α = S∗:(i) For every  > 0, there exists an n ∈ N so that m > n impliesthat am< α + ;(ii) For every  > 0 and every n ∈ N, there exists an m > n sothat am> α − .(c) Honors only! Define the limit superior of {an} as in class:lim supn→∞an= limn→∞supm>nan.Prove that S∗= lim supn→∞an. (Hint: Use part (b)!)Honors Question Chapter #3, Question #23 in