HOMEWORK 5 — DUE OCT 24TH

MATH 25

There are three sections: A,B and C (plus a fourth “optional” one which won’t be graded). Please return the three parts separately to the proper CA.

A. Problems graded by Jennifer

A.1. The AM-GM inequality. Let a_{1}, ⋯, a_{n} be positive real numbers. The arithmetic mean of the a_{k}’s is (a_{1} + ⋯ + a_{n})/n and their geometric mean is \sqrt[n]{a_{1} ⋯ a_{n}}. The AM-GM inequality says that

\sqrt[n]{a_{1} ⋯ a_{n}} ≤ \frac{a_{1} + ⋯ + a_{n}}{n} .

The goal of this problem is to prove the above inequality:

(1) If a, b ≥ 0, compute (\sqrt{a} + \sqrt{b})^{2} − (\sqrt{a} − \sqrt{b})^{2} and deduce that \sqrt{ab} ≤ (a + b)/2;

(2) Use this to prove that if k ≥ 1 and a_{1}, ⋯, a_{2^{k}} are positive real numbers, then

\sqrt[2^{k}]{a_{1} ⋯ a_{2^{k}}} ≤ \frac{a_{1} + ⋯ + a_{2^{k}}}{2^{k}} .

(3) prove the AM-GM inequality. Hint: given a_{1}, ⋯, a_{n} choose k such that 2^{k} ≥ n and set a_{n+1}, ⋯, a_{2^{k}} = \sqrt[n]{a_{1} ⋯ a_{n}}.

A.2. Let {f_{n}}_{n≥0} : R_{≥0} → R be the sequence of functions defined by f_{0}(x) = 0 and

f_{n+1}(x) = f_{n}(x) + \frac{1}{2}(x − f_{n}^{2}(x)).

(1) Prove that for every x ∈ [0, 1],

0 ≤ \sqrt{x} − f_{n}(x) ≤ \sqrt{x} (1 − \frac{\sqrt{x}}{2})^{n} .

(2) Prove that f_{n} converges uniformly to \sqrt{x} on [0, 1].

B. Problems graded by Benjamin

B.1. Simple convergence and uniform convergence I. Let I = [a, b] be an interval and let {f_{n}}_{n≥1} and f be functions from I to R. We say that the sequence {f_{n}}_{n≥1}

• converges simply to f if for every x ∈ I, we have f_{n}(x) → f(x).

• converges uniformly to f if sup_{x∈I} |f_{n}(x) − f(x)| → 0 as n → +∞.

(1) Prove that if {f_{n}}_{n≥1} converges uniformly to f then it converges simply to f;

(2) prove that the converse is not true as follows: let I = [0, 1] and define

f_{n}(x) = x^{n} and f(x) = \begin{cases}0 & if x ≠ 1 \\ 1 & if x = 1. \end{cases}

Show that {f_{n}}_{n≥1} converges simply to f but that the convergence is not uniform.

(3) prove that even if f is continuous then simple convergence does not necessarily imply uniform convergence, by considering

f_{n}(x) = \begin{cases}nx & if 0 ≤ x ≤ 1/(2n) \\ 1 − nx & if 1/(2n) ≤ x ≤ 1/n \\ 0 & if 1/n ≤ x ≤ 1. \end{cases}

B.2. Simple convergence and uniform convergence II. Assume that {f_{n} : I → R}_{n≥1} is a sequence of continuous functions which converges simply to f : I → R, where f is also continuous, and that in addition, f_{n+1}(x) ≥ f_{n}(x) for every n ≥ 1.

(1) show that the sequence {sup_{x∈I} |f_{n}(x) − f(x)|}_{n≥1} is non-increasing;

(2) prove that {f_{n}}_{n≥1} converges uniformly to f.

B.3. Simple convergence and uniform convergence III. Assume that {f_{n} : I → R}_{n≥1} is a sequence of functions which converges simply to f : I → R, and that in addition, for every x ∈ I and every sequence {x_{n}}_{n≥1} which converges to x, we have f_{n}(x_{n}) → f(x)

(1) prove that for every ϵ > 0, there exists δ > 0 and N such that |x − y| < δ implies that |f_{n}(x) − f_{n}(y)| < ϵ for all n > N;

(2) prove that f is continuous;

(3) prove that {f_{n}}_{n≥1} converges uniformly to f.

C. Problems graded by Inna

C.1. Homeomorphisms of [0, 1]. Let f : [0, 1] → [0, 1] be a continuous bijection. Prove that either f(0) = 0 and f(1) = 1 and f is strictly increasing, or f(0) = 1 and f(1) = 0 and f is strictly decreasing.

C.2. Let f : [0, 1] → [0, 1] be a function such that for every x ∈ [0, 1], there exists n_{x} ≥ 1 with f^{n_{x}}(x) = x (where f^{n} is f iterated n times).

(1) Show that f is a bijection;

(2) assume that f is continuous and that f(0) < 1. Find f.

HOMEWORK 5 — DUE OCT 24TH 3

C.3. Periods of functions. Let f, g be continuous functions from R to R and let α > 0 be such that f(x + 1) = f(x) and g(x + α) = g(x). In (3), you can assume that 1 is the smallest period of f and that α is the smallest period of g.

(1) prove that if α ∈ Q then f + g is periodic;

(2) prove that if h is a continuous function which is periodic with two periods 1 and α ∈ R \ Q, then h is constant;

(3) prove that f + g is a periodic function if and only if α ∈ Q. Hint: if f + g is periodic of period λ, what can you say about the function g(x+λ)−g(x) = −f(x+λ)+f(x)?

D. Supplementary problems

D.1. Ascoli’s theorem. Let E be the space of all continuous functions f : [0, 1] → R and let d(f, g) = sup_{x∈[0,1]} |f(x) − g(x)|. Prove that the following are equivalent for a subset K of E:

(1) K is closed, bounded, and equicontinuous (meaning: that for every ϵ > 0, there exists δ such that |x − y| < δ implies |f(x) − f(y)| < ϵ for all f ∈ K);

(2) K is compact.