FINAL EXAM (SAMPLE)MATH 521 FALL 2013, BRIAN COOKRemarks: The final exam has an allotted time of 2 hours. No books, calculators, or anyother items other than a pen/pencil and paper are to be used during the exam. Do notcheat, as cheaters never win, and will be prosecuted to the fullest extent of the law.There are a total of ten questions, equally weighted. Write your answers in the spaceprovided, and if necessary you may write on the back of the page. Be sure to write youranswers using complete sentences and clearly reference any previous results that are beingemployed. Partial credit may be given, however, the partial result must be clear.(1) • Provide the definition of a Cauchy sequence in a metric space.• Is it true that every Cauchy sequence of real numbers is convergent in R? If not,provide a counter example.• Is it true that every Cauchy sequence of rational numbers is convergent in Q? Ifnot, provide a counter example.• If {xn}∞n=1and {yn}∞n=1are two Cauchy sequences of real numbers, show thatthe sequence {xn+ yn}∞n=1is a Cauchy sequence.(2) • Complete the definition. A function f : [a, b] → R (a, b ∈ R, a < b) is uniformlycontinuous if...• If f is continuous on the interval [a, b], show that it is uniformly continuous on[a, b].(3) Recall that a polynomial is a function of the form p(x) = a0+ a1x + a2x2+ ... + anxnfor some real numbers a0, ..., an. Prove that such a function is continuous for x ∈ R.(4) If f is a real valued function, a fixed point of f is a real number x such that f (x) = x.If g is a continuous function on the interval [0, 2], andR20g(t)dt = 2, show thatg has a fixed point in the interval (0, 2). (HINT: Consider the function G(x) :=Rx0(g(t) − t)dt.)12 MATH 521 FALL 2013, BRIAN COOK(5) Let {fn(x)}∞n=1be a sequence of continuous real valued functions on the interval [0,1].If the sequence of functions converges uniformly to the function f, then we have seenthat f is continuous on [0,1]. Thus the Riemann integralsR10fn(x)dx exist for all n,as does the integralR10f(x)dx. Prove that we havelimn→∞(Z10fn(x)dx) =Z10( limn→∞fn(x))dx(6) In some metric space (X, d), prove that any union of open sets is again an open set.(7) Let X denote the set of real valued functions which are continuous on the interval[0,1]. Define the function d : X2→ R byd(f, g) = supx∈[0,1]|f(x) − g(x)|.Show that d is a metric on X.(8) Provide examples of the following:(a) A metric space with a set E ⊂ X that is non empty and has no limit points.(b) A metric space with a set E ⊂ X such that every point of E is a limit point.(c) A metric space with a proper subset E ⊂ X such that every point of X is a limitpoint of E.(d) A metric space with a proper subset E ⊂ X such that every point of E containsthe all of its limit point AND Ec(the complement of E) also contains all of itslimit points.(e) A metric space with a proper subset E ⊂ X such that E has exactly two limitpoints.(9) (a) Provide an example of a series of real numbers which converges, but does notconverge absolutely.(b) Call your underlying sequence {an}n= 1∞. Set f(x) =P∞n=1anxn. Prove thatthe radius of convergence for the power series of f is 1.(10) Let f be a real valued function which is differentiable on the open interval (a, b),continuous on [a, b]. Prove by appealing to definitions that if f0(x) = 0 for allx ∈ (a, b) thenZxaf(t)dt = c(x − a)for some real number