Math 521 Analysis I, Spring 2016

Links

• ABOUT THE FINAL EXAM
• the syllabus (a.k.a. “the old course website”)
• PIAZZA
• Roger Cooke's solutions to the problems in Rudin’s book
• Assigned homework problems:
  • Homework 1 |  Homework 2 |  Homework 3 |  Homework 4
  • Homework 5 |  Homework 6 |  Homework 7 |  Homework 8
  • The last homework assignment
• The first midterm: practice problems.
• The second midterm: review questions and practice problems.
• The third midterm: practice problems.
• The graph of $f(x) = \sin x + \frac12\sin 2x+\cdots+\frac 1n \sin nx$.
• The Google 3D grapher will do its best to produce images of the graphs of functions of two variables. You can (and should) adjust the $x$, $y$, $z$ ranges in the bottom right of the image you get.
  
• Graphs of The Weirdly Differentiable Function and Weierstrass’ even weirder continuous and yet nowhere differentiable function. Also, is the Dow Jones ever differentiable??

Homework assignments

Homework 1

Let $F$ be the set $F = \{a+b\sqrt3 \mid a, b\in\Q\}$. Show that $F$ is a field. Is it an ordered field? In this problem you may assume that $\sqrt{3}\in\R$ exists, and that $F$ is a subset of the real numbers. This implies that many (but not all) of the properties of a field that you would have to verify for $F$ follow from the fact that $\R$ has those properties.

For honors students (and anyone else who feels like it): is the set $G = \{a+b\sqrt[3]2 \mid a, b\in\Q\}$ a field?

See some comments on and solutions of these problems

Homework 2

Ch.1: Use the inner product to do Problem 17.

Ch.2: Problems 2, 3, 4 about countable and uncountable sets.

Ch.2: Problem 11 about “metric spaces”

Extra problem. How does an infinite decimal expansion represent a real number? If $d_1$, $d_2$, … is a sequences of digits (i.e. each $d_k$ is 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9), then we can define the corresponding decimal number by setting \[ 0.d_1d_2d_3d_4\ldots = \sup A \] where $A$ is the set of rational numbers obtained by truncating the decimal expansion to any finite number of decimals, i.e. \[ A = \{x_1, x_2, x_3, x_4, \dots\} \] with \[ x_k = 0.d_1d_2\dots d_k = \frac{d_1}{10} + \frac{d_2}{10^2} + \cdots + \frac{d_k}{10^k}. \] Prove that $0.99999999\dots = 1$.

Homework 3

Chapter 2: 5, 6, 8, 9.

Also these questions:

• is $\mathbb Q$ an open subset of $\R$? Is it a closed subset of $\R$? Does $\R\setminus\mathbb Q$ have any interior points?
• is $\mathbb Z$ an open subset of $\R$? Is it a closed subset of $\R$?

Homework 4

1. Let $X=\R^2$ with the (usual) euclidean distance function. Consider the set $C$, which is the graph of $f(x) = \sin\frac\pi x$, i.e. \[ C = \Bigl\{(x,y) : x\ne 0, y=\sin\frac\pi x\Bigr\} \] Find those points $(x, y)$ in the closure of the set $C$ for which $x=0$.
2. Let $C\subset\R^2$ be the set \[ C = \bigl\{(x, y) : 0\lt x\le 1, y=\sin\frac\pi x\bigr\}. \] Find an open covering of $C$ that has no finite subcover.
3. Let $X = \{x\in\mathbb{Q} : 0\le x\le 1\}$, i.e. $X$ consists of all rational numbers in the interval $[0,1]$, and consider the usual distance function $d(x, y) = |x-y|$. Find an open cover of $X$ that has no finite subcover.
4. For each of the following sets say if it is compact or not. Give a short reason. You may quote theorems from the textbook, or from class (if they are from lecture then you must include the statement). All these sets are subsets of the real line or the euclidean plane $\R^2$. In all cases the intended metric is the usual euclidean distance $d(x,y) = |x-y|$.
   1. $E = [0,\infty)$. Is $E$ a closed subset of $\R$?
   2. $E = \{1\} \cup \{ \frac{n}{n+2} : n\in\mathbb{N}\}$
   3. The parabolic segment $E = \{(x, x^2) : |x|\le 1\}$
   4. The set $E= A\cup B$, where
      • $A = \{(0, y) : |y|\le 1\}$ and
      • $B=\{(x, y) : 0\lt x\le 2, y=\cos(\pi/x)\}$
   5. The set $E= A\cup B$, where
      • $A = \{(0, y) : |y|\lt 1\}$ and
      • $B=\{(x, y) : 0\lt x\le 2, y=\cos(\pi/x)\}$
      (not the same question as the previous one—look carefully.)

Homework 5

1. Consider the sequence of real numbers given by $a_n=n^{-1}$.
   1. Show that $a_n\to 0$, directly from the definition.
      Given $\varepsilon\gt0$ choose an integer $N$ such that $N\gt 1/\varepsilon$. Then for all $n\ge N$ one has $0\lt a_n \le 1/N \lt \varepsilon$.
   2. Show that if a sequence $x_n$ converges, then any subsequence of $x_n$ also converges and has the same limit.
      Let $y_k = x_{n_k}$ be a subsequence of $x_n$, and let $L$ be the limit of $x_n$. In particular this implies that $n_k\ge k$ for all $k$. Given $\varepsilon\gt0$ we choose $N\in\N$ such that $|x_n-L|\lt \varepsilon$ for all $n\ge N$. Then, if $k\ge N$ we have $n_k\ge k\ge N$ and thus $|y_k-L| = |x_{n_k}-L| \lt \varepsilon$.
   3. Show that $b_n=n^{-2}\to 0$ without using the definition of convergence.
      $b_n = a_{n^2}$, so $b_n$ is a subsequence of the sequence $a_n=1/n$. Therefore it converges, and it has the same limit: $\lim_{n\to\infty} 1/n^2 = 0$.
2. Let $x_n$ be a convergent sequence of points in a metric space $(X,d)$. Show that the sequence is bounded.
   Let $x_n\to p$, i.e. let $p$ be the limit of the sequence $x_n$. Choose $\varepsilon=1$. There is an $N\in\N$ such that for all $n\ge N$ on has $d(x_n, p)\lt \varepsilon=1$. Thus, if \[ R=\max\left\{1, d(x_1, p), \dots, d(x_{N-1}, p)\right\}, \] then all $x_k$ satisfy $d(x_k, p)\le R$.
3. Let $x_n$ be a sequence of points in a metric space $(X,d)$. Show that $x_n\to x$ holds if and only if $d(x_n, x)\to0$.
   Consider the sequence of real numbers $a_n = d(x_n, p)$. We are asked to show that $a_n\to0 \iff x_n\to p$.

   Assume $a_n\to 0$. To show $x_n\to p$, let $\varepsilon\gt 0$ be given. By assumption there is an $N\in\N$ such that $|a_n|\lt \varepsilon$ for all $n\ge N$. This implies $d(x_n, p)\lt \varepsilon$ for all $n\ge N$. Therefore $x_n\to p$.

   Assume $x_n\to p$. To show $a_n\to 0$, let $\varepsilon\gt 0$ be given. By assumption there is an $N\in\N$ such that $d(x_n, p)\lt \varepsilon$ for all $n\ge N$. This implies $|a_n|\lt \varepsilon$ for all $n\ge N$. Therefore $a_n\to 0$.

4. Let $(X,d)$ be a metric space, $C\subset X$ a closed subset, and $x_n\in C$ a sequence of points with $x_n\to x$. Show that $x\in C$.
   In order to reach a contradiction, assume that $x\not\in C$. The complement $C^c$ of $C$ is open in $X$, so there is an $\varepsilon\gt0$ such that $B_\varepsilon(x)\subset C^c$. Since $x_n\to x$, there is an $N_\varepsilon\in\N$ for such that $x_n\in B_\varepsilon(x)$ for all $n\ge N$. But then we would have $x_n\not\in C$ for $n\ge N$, which is a contradiction. We conclude that $x\in C$ after all.
5. Let $x_n\in\R^k$ and $y_n\in \R^k$ be two sequences of vectors for which $x_n\to x$ and $y_n\to y$. Show that $(x_n, y_n) \to (x,y)$. Here $(x,y)$ is the dot product, or inner product of the two vectors. In Rudin’s notation (page 16): the assignment is to show that $x_n\cdot y_n \to x\cdot y$.
   The sequences $x_n\in\R^k$ and $y_n\in\R^k$ converge, so they are bounded, i.e. there is an $R\gt0$ such that $\|x_n\|\le R$, and $\|y_n\|\le R$ for all $n\in\N$. We also have \begin{align*} \left|x_n\cdot y_n - x\cdot y\right| &=\left| x_n\cdot y_n - x_n\cdot y + x_n\cdot y - x\cdot y\right| \\ &\le \left| x_n\cdot y_n - x_n\cdot y\right| + \left|x_n\cdot y - x\cdot y\right| & \text{triangle }\le\\ &\le \left| x_n\cdot (y_n - y)\right| + \left|(x_n - x)\cdot y\right| \\ &\le \| x_n\|\,\| y_n - y\| + \|x_n - x\|\,\| y\| & \text{Cauchy }\le \\ &\le R\| y_n - y\| + R \|x_n - x\| \end{align*} Since $x_n$ and $y_n$ converge we have \[ \lim_{n\to\infty} R\| y_n - y\| + R \|x_n - x\| = 0 \] and hence \[ \lim_{n\to\infty} \left|x_n\cdot y_n - x\cdot y\right| =0. \]
6. Let $A\subset\R$, and assume that every term in the sequence $\{x_n\}_{n\in\N}$ is an upper bound for $A$. Show that if $x_n\to x$, then $x$ is also an upper bound for $A$.
   Suppose $x$ is not an upper bound for $A$. Then there is a number $a\in A$ with $a\gt X$. Let $\varepsilon= a-x$. Since $x_n\to X$, there is an $N$ such that for all $n\ge N$ one has $|x_n-x|\lt \varepsilon$. This implies, for $n\ge N$, that $x_n \lt x+\varepsilon = a$. Hence $x_n$ is not an upper bound for $A$ if $n\ge N$, contradicting what was given.

Homework 6

1. Let $x_n=\cos(\theta + n\frac\pi3)$ where $\theta\in(0, \frac\pi3)$ is some constant. For this sequence, compute
   1. $\sup_{n\in\N} x_n$
   2. all possible limits of subsequences of $x_n$
   3. $\limsup_{n\to\infty} x_n$
2. Let $(X,d)$ be a metric space, and let $x_n$ be a convergent sequence in $X$. Show that $x_n$ also is a Cauchy sequence.
3. Give an example of a metric space $(X,d)$ with a Cauchy sequence that does not converge.
4. Give an example of a complete metric space that is not compact. Provide a brief explanation for your answer.
5. Let $x_n$ and $y_n$ be two bounded sequences of real numbers. Show that \[ \limsup_{n\to\infty} (x_n+y_n) \le \limsup_{n\to\infty} x_n + \limsup_{n\to\infty} y_n. \]
6. Find two sequences $x_n$, $y_n$, of real numbers for which \[ \limsup_{n\to\infty} (x_n+y_n) \lt \limsup_{n\to\infty} x_n + \limsup_{n\to\infty} y_n. \]
7. If $x_n$, and $y_n$ are bounded sequences of real numbers for which $x_n$ converges, then show that \[ \limsup_{n\to\infty} (x_n+y_n) = \lim_{n\to\infty} x_n + \limsup_{n\to\infty} y_n. \]

Homework 7

1. Suppose $f:\R\to\R$ is a function that satisfies \[ f(x) = \lim_{h\to 0} \frac12\bigl\{ f(x+h)+f(x-h) \bigr\} \] for all $x\in\R$. Prove or give a counterexample : $f$ is continuous at every $x\in\R$
2. Suppose $f : \R^2\to\R$ is a function for which \[ \lim_{(x,y)\to(0,0)} f(x,y) = L. \] Then show that \[ \lim_{t\to0} f(ta, tb) = L \] for all $(a,b)$, i.e. the limit of $f(x,y)$ as $(x,y)$ approches the origin along the line in the direction $(a,b)$, is $L$ for every $(a,b)$.
3. The following functions are all defined on $\R^2\setminus\{(0,0)\}$. Which of these functions has a limit at the origin? I.e. for which of these functions does $\lim_{(x,y)\to(0,0)}f(x,y)$ exist?
   1. $\displaystyle f(x, y) = \frac{x} {x^2+y^2}$
   2. $\displaystyle f(x, y) = \frac{x^2} {x^2+y^2}$
   3. $\displaystyle f(x, y) = \frac{xy} {x^2+y^2}$
   4. $\displaystyle f(x, y) = \frac{xy} {x^2+y^4}$
4. Conjecture: Suppose $f : \R^2\to\R$ is a function for which \[ \lim_{t\to0} f(ta, tb) = L \] for all $(a,b)$ (i.e. the limit of $f(x,y)$ as $(x,y)$ approches the origin along the line in the direction $(a,b)$, is $L$ for every $(a,b)$). Then \[ \lim_{(x,y)\to(0,0)} f(x,y) = L. \] Is the conjecture true or false? (Consider the examples from the previous problem).
5. If $f:\R\to\R$ is any function, then the graph of $f$ is by definition the set \[ G_f = \bigl\{(x,y) \in\R^2 : y=f(x)\bigr\}. \] Prove or give a counterexample for each of the following two statements:
   1. If $f$ is continuous then $G_f$ is a closed subset of $\R^2$.
   2. If $G_f$ is a closed subset of $\R^2$, then the function $f$ is continuous.

Homework 8

• The Weirdly Differentiable Function See if you can zoom in and find this portion of the graph:
  
• Weierstrass’ even weirder continuous and yet nowhere differentiable function.
• Also, is the Dow Jones ever differentiable??

1. Which of the following functions are uniformly continuous on the stated domains? If a theorem on uniform continuity applies, you may of course use it. Problem 3 below may also be useful in some cases.
   1. $f:\R\to\R$, $f(x) = e^{-x^2}$
   2. $f:[-10,10]\to\R$, $f(x) = e^{x^2}$
   3. $f:\R\to\R$, $f(x) = e^{x^2}$
   4. $f:\R\to\R$, $f(x) = \bigl(1+x^2\bigr)^{-1}$
   5. $f:\R\to\R$, $f(x) = 1+x^2$
   6. $f:\R\to\R$, $f(x) = e^x$
2. Let $f:\R\to\R$ be a uniformly continuous function. Show that there is a constant $C\gt 0$ such that $f(x) \le Cx$ for all $x\ge 1$.
3. Let $f:I\to\R$ be a differentiable function on some interval $I\subset \R$. Assume that there is a number $M\in\R$ such that $|f'(x)| \leq M$ for all $x\in I$. Show that $f$ is uniformly continuous.(Hint: use the Mean Value Theorem.)
4. Let $I\subset \R$ be some interval. A function $f:I\to\R$ is called Hölder continuous of order $\alpha$ if there is a number $C$ such that $|f(x)-f(y)|\le C |x-y|^\alpha$ for all $x, y\in I$.
   1. Show that every Hölder continuous function is uniformly continuous.
   2. Find a function $f:[-1,1]\to\R$ which is Hölder continuous of order $\alpha=1/3$ but not of order $\beta=2/3$.
   3. Consider the function \[ f(x) = \frac{-1}{\ln x} \] on the interval $[0, \tfrac12]$. Is $f$ Hölder continuous for any $\alpha\gt0$? Is $f$ uniformly continuous?
5. Consider the function whose graph is shown above: \[ f(x) = \begin{cases} x + \frac12 x^2\sin\frac\pi x & x\neq0\\ 0 & x=0 \end{cases} \]
   1. Show that $f$ is differentiable at $x=0$
   2. Find a sequence $x_n\in\R$ with $x_n\to0$ and $f'(x_n) \lt 0$
6. Let $I\subset\R$ be some interval, and let $f:I\to\R$ be a function which has derivatives of all orders $\le n+1$. Show that for any $a, x\in I$ there is a number $t$ between $a$ and $x$ such that \begin{multline} f(x) = f(a) + (x-a) f'(a) + \frac{1}{2!}(x-a)^2 f''(a) + \cdots + \frac{1}{n!} (x-a)^nf^{(n)}(a)\\ + \frac{1}{n!}(x-a)(x-t)^{n} f^{(n+1)}(t). \end{multline} Hint: apply the Mean Value Theorem on the interval $a\le t\le x$ to the function \[ g(t) = f(t) + (x-t) f'(t) + \frac{1}{2!}(x-t)^2 f''(t) + \cdots + \frac{1}{n!} (x-t)^nf^{(n)}(t). \]
7. Newton’s binomial theorem.  Use the result from the previous problem to show that for every $x\in[0,1)$ one has \[ (1-x)^\alpha = 1 -\alpha x+ \frac{\alpha(\alpha-1)}{2!}x^2 - \frac{\alpha(\alpha-1)(\alpha-2)}{3!} x^3+ \cdots \;. \] In other words show that the Taylor–Mclaurin series for $f(x) = (1-x)^\alpha$ converges to $f(x)$ for every $x\in[0, 1)$.
   For this week’s homework it is enough to do this problem for $\alpha=-1$

Homework 9

1. Show that the function \[ f(x) = \begin{cases} 0 & 0\leq x\lt 1 \\ 1 & 1\leq x\leq 2 \end{cases} \] is Riemann integrable on the interval $[0,2]$.
2. Let $f:[a, b]\to\R$ be a nondecreasing function. Show that if $P$ is a partition of $[a,b]$ into $N$ pieces of equal length (being $\Delta x= (b-a)/N$), then \[ U(f, P) - L(f, P) = \frac{1}{N}\bigl(f(b)-f(a)\bigr)(b-a). \] Show that $f$ is Riemann integrable.
3. Let $f:\R\to\R$ be the function defined by $f(x) = 0$ for all $x\neq 0$, and $f(0) = 1$. Show that $f$ is Riemann integrable on the interval $[-1, 1]$ and prove from the definition that $\int_{-1}^1 f(x) dx = 0$.
4. Let $X$ be the set of all continuous functions $f:[0, 2\pi]\to \R$ ($X$ is usually written as $C([0, 2\pi])$). Let \[ d(f, g) = \sup_{0\leq x\leq 2\pi} |f(x)-g(x)| \] be the standard distance function on $X$.
   1. Consider the functions $f_n(x) = \sin (2^n x)$ where $n\geq0$ is an integer. Show that $d(f_n, f_m) \geq 1$ whenever $n\neq m$.
   2. Is the set $A = \{f_n : n=0,1,2,3,\dots\}$ a closed subset of $X$?
   3. Is the set $A$ a bounded subset of $X$?
   4. Is the set $A$ compact?