MA104 EXAM IProblem 1:For (xn) a b ounded sequence of real numbers, and L ∈ R a particular real number,consider the statement:(A) := “For all ε > 0 the set {n : xn∈ (L − ε, L + ε)} is infinite.”It is not equivalent to the statement:(B) := “xnconverges to L.”Now consider the following two statements. One is true and the other is false.(i) (A) ⇒ (B)(ii) (B) ⇒ (A)Determine which one is which. Then prove the true statement, and find a coun-terexample for the false statement.Problem 2:For any nonempty bounded s ubset E ⊂ R of the real numbers, define the diameterof E:diam(E) := sup{|x − y| : x, y ∈ E}For a given real sequence (xn), define the setTk:= {xn: n ≥ k}.(One might call this the kthtail of the sequence.)Show that (xn) is convergent if and only if diam(Tk) converges to 0.Problem 3:Show the following inequalities hold for any two bounded real nonnegative sequences(xn) and (yn). Also give an example to show a strict inequality is possible in eachcase.(I) lim sup(xn· yn) ≤ (lim sup xn) · (lim sup yn)(II) lim sup(xn+ yn) ≤ lim sup xn+ lim sup yn12Problem 4:For any fixed real number a ∈ R, the sequence (an) may do one of several things.Fully characterize the possibilities, and prove your assertions. In the cases where(an) diverges, specify also the nature of the divergence.Problem 5:For any two nonempty bounded subsets E, F ⊂ R, define the distance from E toF bydist(E; F ) := inf{|e − f | : e ∈ E, f ∈ F }Indicate whether each of the following statements is true or false. If true, thenprove; if false, give a sp e cific counterexample. (E, F, G denote arbitrary nonemptybounded sets in R.)(A) dist(E; F ) = 0 ⇐⇒ E = F(B) dist(E; F ) ≤ dist(E; G) + dist(F ; G)(C) dist(E; F ) = 0 ⇒ inf(E) ≤ sup(F )(D) sup(E) ≥ inf(F ) ⇒ dist(E; F ) = 0Problem 6:Let X denote the set of all “0, 1 sequences.” In other words, elements of X aresequences, all of whose terms are 0 or 1. An equivalent formal definition for X isthe set of all functions x : N → {0, 1}. For (xn), (yn) ∈ X defined((xn), (yn)) :=∞Xn=1|xn− yn|2n.Show in detail that d is a metric, and furthermore that X is complete with respectto this metric.Hint(s for all the problems): Recall that for a given sequence snwe say that theseriesPsnconverges if the sequence of partial sums Σn:= s1+ ... + snconverges.If you are aware of results from homework problems (like a formula that mighthelp in problem 6), or facts that we use repeatedly in lecture, feel free to cite themusing our familiar phrases and abbreviations. Also reme mber to precisely statewhat it is you are proving, and where applicable, what you are not proving.If you cannot recall what a metric is, then maybe the phrases ‘positive defi-nite,’ ‘symmetric,’ and ‘triangle inequality’ will help. Also remember the standardabsolute value is itself a metric.Sometimes it helps to immediately make a cheat sheet of all of the more cumber-some definitions and theorems. If you do this, you may refer to it in your solutionsand submit it as part of your exam. No c redit will be given in any case for simplyciting facts, without indicating relevance to the problem.I won’t take any questions about the exam during the exam. If you have toleave, give me your work first, and return as quickly as possible. If you have readall this, stare at the board to my