MATH 210 Final, Fall 2017

MATH 210: Introduction to Analysis Fall 2015-2016, Final, Duration: 120 min. Exercise 1. The contraction mapping theorem. Let f : R → R be a differentiable function. Assume that there is 0 < k < 1 such that |f′(x)| ≤ k < 1 for all x ∈ R. Let x0 ∈ R be any real number and define a sequence by setting xn+1 = f(xn). (a) (5 points) Prove that f is a contraction. (b) (10 points) Recall the definition of a Cauchy sequence. (c) (10 points) Prove that the sequence {xn} is Cauchy. (d) (5 points) Deduce that the sequence {xn} converges to a real number x∗. (e) (5 points) Prove that x∗ is a fixed point of f. (f) (5 points) Prove that x∗ is the unique fixed point of f. Exercise 2. (a) (10 points) Let (X, d) be a metric space and let f : X → R be function. State the definition of continuity of f at x0. (b) Let (X, d) be a metric space and let A ⊂ X be dense in X. Assume that f, g : X → R are two continuous functions such that f(x) = g(x) for all x ∈ A. The goal of this question is to prove that f and g are equal. Fix x0 ∈ X \ A and ε > 0. i. (5 points) Find δ such that if d(x, x0) < δ then |f(x) − f(x0)| < ε and |g(x) − g(x0)| < ε. ii. (3 points) Show that for any y ∈ X, |f(x0) − g(x0)| ≤ |f(x0) − f(y)| + |f(y) − g(y)| + |g(y) − g(x0)|. iii. (10 points) Show using the previous question that |f(x0) − g(x0)| ≤ 2ε. iv. (5 points) Deduce that f(x0) = g(x0). v. (2 points) Deduce f and g are equal. (c) (5 points) Let f : R → R be a continuous function such that f(x) = 0 for all x ∈ Q. Determine f. Exercise 3. (20 points) Let n ≥ 2 be an integer. Using lower and upper sums for an appropriate partition prove that 1 2 + 1 3 + · · · + 1 n ≤ log n ≤ 1 + 1 2 + 1 3 + · · · + 1 n − 1. Exercise 4. (a) (10 points) State Rolle’s Theorem. (b) Consider the function f(x) = � x+x3 0 (1 + sin2t)dt. i. (5 points) Find a root of f. ii. (10 points) Show using Rolle’s theorem that f has only one root on R. Exercise 5. Let fn : [0, 1] → R defined by fn(x) :=        n2x(1 − nx) if x ∈ [0, 1 n] 0 otherwise. (a) (10 points) Study the pointwise convergence of fn on [0, 1]. (b) (10 points) After justifying that fn is Riemann integrable, compute � 1 0 fn(t)dt for all n ≥ 1. 2 (c) (10 points) Study the uniform convergence of fn on [0, 1]. (d) (5 points) Let 0 < a < 1. Study the uniform convergence of fn on [a, 1]. Exercise 6. The goal of this exercise is to give a new proof of the convergence of the series � 1 nα , when α > 1. Let α > 1 and consider the following function on (0, +∞) defined by f(x) := − 1 (α − 1)xα−1 . (a) (5 points) Compute f′. (b) (5 points) Apply the Mean Value Theorem to f on the interval [k, k + 1]. We denote by xk ∈ (k, k + 1) the point obtained by this theorem. (c) (10 points) Using the previous question, prove that n � k=1 1 (xk)α = 1 α − 1 − 1 (α − 1)(n + 1)α−1 . (d) (5 points) Prove that n � k=1 1 (k + 1)α < n � k=1 1 (xk)α . (e) (5 points) Deduce that the sequence n � k=1 1 kα is bounded. (f) (5 points) Deduce that � 1 nα is convergent. Exercise 7. Let f : R → R be a differentiable function. (a) (10 points) Supose that f is twice differentiable at x0. Prove that lim h→0 f(x0 + h) − 2f(x0) + f(x0 − h) h2 = f′′(x0). (b) (5 points) Assume that the limit of f(x0 + h) − 2f(x0) + f(x0 − h) h2 as h → 0 exists. Disprove via a counterexample that f is twice differentiable at x0.