# Math 314, Spring 2013

Note: this is an archived version of the course page. The formatting has been simplified and student comments have been erased.

## Welcome

In this class we will begin to learn about real analysis—the field of mathematics that ultimately provides the foundation for calculus. The course begins with a rigorous treatment of the real number system. We will also study the topology of the real numbers, which means we will think of it as a space where the idea of continuity makes sense. We will conclude with the very beginnings of calculus.

## Course information

Course title: Foundations of analysis
Meeting times: T,Th 12–1:15pm
Meeting place: ILC-401
Instructor office: MG 237-A
Office hours: M 9am, and Th 4:30pm
Textbook: Understanding Analysis, by Stephen Abbott

## Topics

In this class we will learn about a whole lot of “C”-words:

• Construction the real number system
• Cardinality
• Completeness
• Convergent and Cauchy sequences
• Continuity
• Compactness
• Connectedness

Written work and attendance: This will include Weekly homework assignments as well as some in-class worksheets. Attendance and participation are counted as well. Together this will be worth about one fourth of your grade. I will drop your lowest homework grade.

Exams: There will be two midterms and a non-cumulative final, each worth about one fourth of your grade. The tentative exam dates are as follows:

## Homework

HW12 (due Thursday, May 9)
§ 5.2. Exercise 2ab.

Problem A. Use the definition directly to find the derivatives of the following functions:

• $x^3$
• $1/\sqrt{x}$

Problem B. Determine where each of the following functions is differentiable. Justify your answers.

• $|x|+|x-1|$
• $x\cdot|x|$
• $|\sin(x)|$

§ 8.2. Exercises 2 and 6. (Hint for #6: use inequality 7.4.2(v).)

Problem C. Find the definitions of open and closed subset of a metric space on page 224. Prove that a set $A$ is open iff the complement $A^c$ is closed.

HW11 (due Tuesday, April 30)
§ 4.5. Exercise 2.

Problem A. Suppose that $f(x)$ is continuous and one-to-one on the interval $[a,b]$. Show that $f(x)$ is either always increasing or always decreasing.

§ 3.5. Exercise 8.

Problem B. Show how to write each of the sets as a union of countably many closed sets. Also, decide whether each set is meager or non-meager.

• $(0,1)$
• $[0,1)$
• $\mathbb{Z}$
• $\mathbb{Q}$
• $\mathbb{R}\smallsetminus\mathbb{Z}$

HW10 (due Tuesday, April 23)
§4.3. Exercises 1 and 7.

Problem A. Decide for which points $c$ each of the following functions is continuous at $c$.

• $f(x)=\frac{x^2+2x+3}{x^2+1}$
• $f(x)=\frac{x^2+x+3}{x^2-1}$

Problem B. Give an example of a discontinuous function $f$ such that $|f|$ is continuous. Give an example of discontinuous functions $f$ and $g$ such that $f+g$ is continuous.

§4.4. Exercises 4 and 6.

Problem C. Suppose that $f$ is continuous on $[0,1]$, and that for every $n$ there exists $x\in[0,1]$ such that $|f(x)|\leq\frac1n$. Show that there exists $x\in[0,1]$ such that $f(x)=0$.

Problem D. Prove that $\sqrt{x}$ is uniformly continuous on $[1,\infty)$. Using the fact that $[0,1]$ is compact, show how to conclude that $\sqrt{x}$ is uniformly continuous on its domain $[0,\infty)$.

HW9 (due Tuesday, April 16)
§4.1. Read the section and write a short response, guided by the following questions. Before reading the section, how would you have described what it mean for a function to be continuous? How about now? In what ways was your old description good? In what ways was it inadequate? How do examples and counterexamples guide your thinking?

§4.2. Exercises 2 and 6.

Problem A. Determine a condition on $|x-4|$ that will ensure that $|\sqrt{x}-2|<\frac12$. Determine a condition on $|x-4|$ that will ensure that $|\sqrt{x}-2|<\frac{1}{100}$. Prove using Definition 4.2.1 that $\lim_{x\to 4}\sqrt{x}=2$.

Problem B. Prove that $\lim_{x\to2}\frac{1}{1-x}=-1$.

HW8 (due Tuesday, April 2)
§3.3. Exercise 4.

Problem A. Decide whether the following sets are compact. For those that are not compact, provide (i) a sequence in the set which has no subsequential limit in the set; and (ii) an open covering of the set with no finite subcovering.

1. $\ZZ$
2. $\{1,2,3,4,5\}$
3. $\set{q\in\mathbb Q\mid 0\leq q\leq2}$
4. $\set{1/n\mid n\in\mathbb N}$
5. $\set{1/n\mid n\in\mathbb N}\cup\{0\}$

Problem B. Prove, using the “subsequential limit” definition of compactness, that if $A$ and $B$ are compact, then $A\cup B$ is compact.

§3.4. Exercise 7.

Problem C. Show that every nonempty open set contains a perfect set.

Problem D. Suppose that $I_1$ and $I_2$ are open intervals. Under what scenario will $I_1\cup I_2$ be connected? Prove you are right.

HW7 (due Tuesday, March 19)
§3.2. Exercises 3 and 7.

Problem A. Prove or give a counterexample: $\overline{A\cap B}=\overline{A}\cap\overline{B}$.

Problem B. Suppose that $D\subset\mathbb R$ and that $D$ is dense, that is: for every $a<b$ there exists $d\in D$ such that $a<d<b$. Show that if $D$ is closed, then $D=\mathbb R$.

HW6 (due Thursday, March 14)
§2.6. Exercise 1.

Problem A. Show directly from the definition that $\frac{n+1}{n}$ is Cauchy.

Problem B. We have seen that the sequence of partial sums of the harmonic series $S_n$ is divergent even though it satisfies $\lim(S_{n+1}-S_n)=0$. Find another example of a divergent sequence $a_n$ which satisfies $\lim(a_{n+1}-a_n)=0$.

§2.7. Exercises 4 and 5(a).

Problem C. Show that the series converges:
$1-\frac12-\frac13+\frac14+\frac15-\frac16-\frac17++–\cdots$

Problem D. Show that $\sum\frac{1}{n(n+1)(n+2)}=\frac14$. [Hint: telescope!]

Problem E. Suppose $a_n>0$ for all $n$, and let $b_n=\frac{a_1+\cdots+a_n}{n}$. Show that $\sum b_n$ diverges.

HW5 (due Tuesday, March 5)
§2.4. Exercise 5.

Problem A. Show that the sequence $\sqrt{2},\sqrt{2+\sqrt{2}},\sqrt{2+\sqrt{2+\sqrt{2}}},\ldots$ converges and find its limit.

Problem B. Determine whether the series converge or diverge:

• $\sum\frac{1}{n\ln(n)}$
• $\sum\frac{1}{n\ln(n)^2}$

Problem C. Show that $\int_1^\infty\frac1{x^p}$ converges if and only if $p>1$.

§2.5. Exercise 3.

Problem D. Use the divergence criterion to show the sequence diverges: $x_n=\sin(n\pi/4)$

Problem E. Suppose that $x_n$ has the property: every subsequence has a further subsequence which converges to $L$. Show that $x_n\to L$. [Hint: try contradiction!]

HW4 (due Thursday, February 21)
§2.3. Exercises 2, 4, 7a, 8, 9

Problem A: Show that if $\lim a_n=a$ and $a\neq0$ and $\lim a_nb_n=ab$ then $\lim b_n=b$.

Problem B: Use the algebraic limit theorems to evaluate the limits:

• $\displaystyle\lim\left(2+\frac1n\right)^2$
• $\displaystyle\lim\frac{\sqrt{n^2+1}}{n-2}$

HW3 (due Thursday, February 14)
§2.1. Read the section and write a several-paragraph response. Here are some questions to guide your writing. Does it ever make sense to add together infinitely many numbers? Does it always? Do the usual rules of addition still apply in the case when there are infinitely many summands? How can mathematicians be sure they aren’t making mistakes by assuming everything works as expected? Did you have any questions while reading this section?

§2.2. Exercises 1, 2, 4, 6, 7, and 8.

HW2 (due Thursday, February 7)
This homework covers three sections, so I strongly recommend that you get started over the weekend and come in on Tuesday to ask questions!
§1.3. Exercises 2, 3ab, 4, and 7
§1.4. Exercises 2, 3, 4, 5, and 10
§1.5. Exercises 3, 4, 5, 6, and 8

HW1 (due Tuesday, January 29th)
§1.1. Read the section and write a response (a few paragraphs) discussing the following questions.

• What are numbers and what kinds of numbers are there?
• In what ways are the rational numbers satisfactory and unsatisfactory as a number system?