FINAL EXAM (SAMPLE) MATH 521 FALL 2013, BRIAN COOK Remarks: The final exam has an allotted time of 2 hours. No books, calculators, or any other items other than a pen/pencil and paper are to be used during the exam. Do not cheat, 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 space provided, and if necessary you may write on the back of the page. Be sure to write your answers using complete sentences and clearly reference any previous results that are being employed. 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? If not, provide a counter example. ∞ • If {xn }∞ n=1 and {yn }n=1 are two Cauchy sequences of real numbers, show that the sequence {xn + yn }∞ n=1 is a Cauchy sequence. (2) • Complete the definition. A function f : [a, b] → R (a, b ∈ R, a < b) is uniformly continuous 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 + a1 x + a2 x2 + ... + an xn for 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 R 2 x such that f (x) = x. If g is a continuous function on the interval [0, 2], and 0 g(t)dt = 2, show that gR has a fixed point in the interval (0, 2). (HINT: Consider the function G(x) := x (g(t) − t)dt.) 0 1 2 MATH 521 FALL 2013, BRIAN COOK (5) Let {fn (x)}∞ n=1 be a sequence of continuous real valued functions on the interval [0,1]. If the sequence of functions converges uniformly to the function R 1 f , then we have seen that f is continuous on [0,1]. Thus the Riemann integrals 0 fn (x)dx exist for all n, R1 as does the integral 0 f (x)dx. Prove that we have Z 1 Z 1 lim ( fn (x)dx) = ( lim fn (x))dx n→∞ 0 0 n→∞ (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 : X 2 → R by d(f, g) = sup |f (x) − g(x)|. x∈[0,1] 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 limit point of E. (d) A metric space with a proper subset E ⊂ X such that every point of E contains the all of its limit point AND E c (the complement of E) also contains all of its limit points. (e) A metric space with a proper subset E ⊂ X such that E has exactly two limit points. (9) (a) Provide an example of a series of real numbers which converges, but does not converge absolutely. P n (b) Call your underlying sequence {an }n = 1∞ . Set f (x) = ∞ n=1 an x . Prove that the 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 f 0 (x) = 0 for all x ∈ (a, b) then Z x f (t)dt = c(x − a) a for some real number c.