# Math 413 and Math 513 (Phillips): Introduction to Analysis 1

This is the home page for N. C. Phillips' Introduction to Analysis 1 (Math 413 and Math 513) the University of Oregon, Fall quarter 2018.

## Summary of updates (most recent first)

• 14 December 2018:
• Solutions to Homework 9 posted.
• Corrected versions of syllabus and solutions to real and sample midterm and final posted, with difference files. The corrections to the syllabus will be incorporated in next quarter's syllabus.
• 10 November 2018: The pdf version of the syllabus has been updated with the changes already made to the course web page. In particular, the change in homework for weeks 7--9 has now been included there. In addition, there is now a hint there on Problem 3.14(a) in the Week 7 homework, which isn't on the course web page. (It contains several formulas.)
• 6 November 2018:
• Syllabus changed starting with Week 7. Week 7 is now a continuation of Chapter 3, and has a not previously posted homework assignment. All the original homework assignments from Week 7 on have been postponed one week. (The original week 9 assignment is no longer on the course page, but it will be the first assignment for Math 414.)
• Office hours changed. Wednesdays 3--3:00 pm has been moved to Fridays 3--3:00 pm and Thursdays 11--11:50 am has been moved to Thursdays 10--10:50 am.
• Corrected solutions to Homework 4 have been posted (pdf). Solutions to Problems 6 and 10 were missing in the original version.
• (Partial) solutions to Homework 6 have been posted (pdf).
• 16 October 2018: Files newly posted:
• 2 October 2018: Files newly posted:
• 28 September 2018:
• Point values have been assigned to the problems in Homework 1 and Homework 2. This page and the pdf version of the syllabus have been updated. The pdf version of the syllabus also now has comments on Problem 5 on Homework 1, about mistakes seen in the past.
• 24 September 2018:
• The questionnaire (due Wednesday 26 September) was revised about 12:35 pm Monday 24 September. (If you already filled out the old version, just add the newly requested information on the back.)

## Basic information

• Course number: Math 413, CRN 13786, and Math 513, CRN 13794,
• Course title: Introduction to Analysis 1.
• Time and place: MWF 11:00--11:50 am, 260 Condon. Discussion section: Tu 11--11:50 am, 104 Deady.

I will be away at conferences 22--26 October (5th week of classes) and 26--30 November (10th week of classes). I will arrange makeup lectures if possible. More likely, the lectures will be given by substitutes.

• Instructor: N. Christopher Phillips.
• Office: 105 Deady. The office is on the main level at the east end of the building, opposite the stairs (very close to the classroom). Please knock. I can't leave my door open, because if I do I get too many people asking to borrow my telephone or pencil sharpener, or where to find the math department office or nonexistent rooms (such as 350 Deady).
• Office hours: MF 3:00--3:50 pm; Th 10--10:50 am, or by appointment.
• Email. The subject line of your message should start with "M413", followed by your last name, then first initial.

I might use email occasionally to distribute general announcements. I will give short replies to emailed questions. I don't type at a reasonable speed, so I will rarely answer complicated questions by email. Please come to office hours instead.

When emailing me, please use plain text (7 bit ASCII) only. That is, only the characters found on a standard English language keyboard; no curved quotation marks, curved apostrophes, accented letters, Greek letters, etc. Use TeX pseudocode for mathematical symbols, including Greek letters. In particular:

• No html encoded (web page format, or "styled") messages. See "Configuring Mail Clients to Send Plain ASCII Text" for how to turn off html. (The University of Oregon webmail program is apparently capable of sending plain text email.)
• No binary files or attachments (except by prior arrangement).
• No Microsoft Word files. I do not accept these under any circumstances, since I don't have software that reads them.
• No mime encoding or other encoding of ordinary text messages.
• Course description: The three quarter sequence Math 413--Math 415 give a rigorous presentation of calculus of one and several variables (the material of Math 251--Math 253 and Math 281--Math 282). In Math 413, we will basic properties of the real numbers and complex numbers, elementary point set topology and continuity in the context of metric spaces (which includes all subsets of n dimensional real and complex space), convergence of series, and differentiation.
• Textbook: W. Rudin, Principles of Mathematical Analysis 3rd edition.
• Syllabus (pdf); LaTeX. (Corrected and updated version posted 10 November 2018.) Everything here is also there, with better formatting for the mathematical expressions and sometimes in more detail, but without links. Exception: My email address isn't there, since spammer address harvestors read pdf files.
• Notes on Terminology: The book contains a few oddities of notation and terminology which I will not follow. Open intervals are referred to as segments''. The sets of real numbers and rational numbers are called R and Q instead of the more usual blackboard bold (or boldface) versions. (In particular, I will feel free to use the letters R and Q for other purposes.) In contrast to this, the book suddenly switches from R to R when the function is supposed to take values in R^n instead of a general metric space; I will make no such distinction.

See the corresponding part of the pdf version of the syllabus; for more, on the set of natural numbers, on "increasing" vs. "strictly increasing" vs. "nondecreasing", etc.

### Homework

Contribution to final grade: approximately 20%. Due every Monday, except as otherwise stated.

The homework will be hard, some of it quite hard, but it is an essential part of the course.

The homework consists of proofs. This means that everything must be proved. In particular, if a problem asks for an example or counterexample, you must prove that your example has the required properties. If a problem asks whether something is true, you must not only decide whether it is, but also provide a proof or counterexample as appropriate; just as above, counterexamples must be proved to have the required properties.

Proofs should be clear, correct, and complete. They should use complete sentences and correct punctuation. They should have all quantifiers in the right places. They should be in logical order. For example, for a proof of convergence of a sequence from the definition, start with "Let epsilon be greater than zero" and end with the statement that something is less than epsilon. If there are subsidiary arguments in a proof, say what is being claimed, say that you are proving it, prove it, and say when the proof of your claim is done. A reader who can't follow any of the mathematics should nevertheless be able to easily tell where the proof of your subsidiary claim starts and where it ends.

I expect to post on the course website at least sketches of some solutions after the homework is turned in.

Some procedural points:

• Staple pages together. Don't fold or tear corners or fold in half.
• If you cooperate with someone else on any particular problem, you must acknowledge the cooperation (including the name of the person) in your solution.

### One Midterm

Contribution to final grade: approximately 30%. Date: Friday 26 Oct.

The midterm from the last time I taught the course is here (pdf); LaTeX. Its solutions are here (pdf); LaTeX. The format of the midterm this time may be a bit different. A large portion of the midterm will ask for proofs which you will have to devise for yourself during the exam period. If I use the format from last time, many people will not finish, and it is likely that getting 75% of it completely right will be a very good score.

I am willing to allow people to start an hour early or continue an hour late, or to arrange to give the entire exam during a longer period outside of the normal class time. (This was done the last time I taught the course.) Such a procedure can only be followed if suitable rooms and times can be found, and if everybody enrolled in the class agrees to it.

### Final Examination

Contribution to final grade: approximately 50%. Date: Thursday 6 Dec., 10:15 am--12:15 pm, in our usual classroom. (The final exam schedule is here.)

No early final exams, according to University rules. If you have another final exam scheduled at the same time as our final exam, you need to give me the details (course number and instructor) by Monday of the 7th week of classes.

Similar comments on the content apply as for the midterm. The final exam from the last time I taught the course is here (pdf); LaTeX. Its solutions are here (pdf); LaTeX.

I keep originals of the final exam, but will give out on request a scan or copy of your exam.

### Learning Disabilities

Students with documented learning disabilities who wish to use the Accessible Education Center to take tests under specifically arranged conditions should let me know as soon as possible, certainly by Wednesday of the third week of classes. Such students must also be sure to meet the Accessible Education Center's separate deadlines for requests; these are likely to be a week or more in advance of the exam date (much more for final exams). I can't do anything to help a student who misses its deadline. (I have tried in the past.)

Complaints about the grading of any exam must be submitted in writing by the beginning of the first class period after the class in which that exam is returned.

### Extra Credit

Extra credit will be awarded to the first person to detect any error or misprint in the book, in homework solutions, or in exams and exam solutions. More extra credit will be given for catching mathematical errors than misspelled words or wrong dates. To get the extra credit, you must tell me what the correct version should be.

There are likely to be a number of mistakes to catch. When going through materials from the last time I taught this course, I found several misprints, including one egregious one ("not compact" for "compact"), which were not caught despite a similar offer for extra credit.

## Learning Outcomes

The course deals with:
• The basic topology of the real numbers and the complex numbers.
• The topology of metric spaces.
• Convergence of sequences and series of real numbers and complex numbers.
• Continuity of functions and continuous functions between metric spaces.
• Differentiability of functions, differentiable functions, and their derivatives, for functions with domain a suitable subset of the real numbers and codomain the real numbers or the complex numbers.
The successful student will be able to precisely state the definitions associated with these topics, and will be able to rigorously prove standard facts. The successful student will also be able to rigorously prove or disprove (as appropriate) statements about these topics which have not been encountered before, and which are at a level of difficulty appropriate to a mid-level introductory analysis course. The successful student will be able to write these proofs so that they are correct, complete, clear, readable, and in logical order.

The code of student conduct and community standards is here. In this course, it is appropriate to help each other on homework as long as the work you are submitting is your own and you understand it, and you give the names of any people you cooperated with. It is not appropriate to help each other while taking exams, to look at other students' exams while taking exams, or to bring any unauthorized material to exams.

## Schedule

We will cover most of Chapters 1--5 of the book in the fall quarter, and we will follow the book fairly closely.

The book usually used in the preceding course is S. Abbott, {\emph{Understanding Analysis.}} Much of the material of this quarter is also there, but often restricted to special cases. (As a representative example, continuity is mostly only treated there in the context of functions from subsets of the reals to the reals, and we will consider continuity of functions between metric spaces.) One needs to have had some exposure to this material, at least for motivation; otherwise, the first several weeks of this course will seem hopelessly abstract.

The schedule gives approximate times that topics will be discussed. It is necessary to read the book; not everything will be done in class. Homework assignments are subject to change.

### Week 1 (24--28 Sept.)

Chapter 1 (the system of real numbers). Turn in the questionnaire Wednesday 26 September.

Homework (due Monday 1 Oct.): Chapter 1: 1, 2, 5, 6, 9, 13, 17. Point values: 10 points per problem, except Problem 6: 10 points for each part (total 40).

See the pdf version of the syllabus for comments on Problem 6.

### Week 2 (1--5 Oct.)

Start Chapter 2 (metric spaces: basic properties).

Homework (due Monday 8 Oct.): Chapter 2: 2, 3, 4, 5, 6, 8, 9, 11. Point values: 5 points for each part of each problem. For this purpose, Problem 6 is considered to have three parts (it asks you to do three things, even though they are not officially labelled as parts), and similarly Problem 8 is considered to have two parts.

### Week 3 (8--12 Oct.)

Finish Chapter 2 (more on metric spaces: compactness, etc.).

Homework (due Monday 15 Oct.): Chapter 2: 14, 16, 19, 20, 22, 23, 25. Point values: 9 points for each part of each problem, except 10 points for Problem 23. For this purpose, Problem 16 is considered to have two parts (it asks you to do two things, even though they are not officially labelled as parts), and similarly Problem 25 is considered to have two parts.

### Week 4 (15--19 Oct.)

Start Chapter 3 (sequences).

Homework (due Monday 22 Oct.): Chapter 3: 1, 2, 3, 4, 5, 6, 10, 21, 22. Point values: 10 points per problem, except 20 points for Problem 6 (5 points for each of its parts).

### Week 5 (22--26 Oct.)

Continue Chapter 3 (series); review; midterm. (The midterm is Friday 26 Oct.)

Homework (due Monday 29 Oct.; short assignment): Chapter 3: 7, 8. Point values: 10 points each.

### Week 6 (29 Oct.--2 Nov.)

Continue Chapter 3 (series).

Homework (due Monday 5 Nov.): Chapter 3: 9, 16, 23, 24. Also do Problem A on the pdf version of the syllabus. (It asks you to prove the equivalence of four definitions of the lim sup of a sequence.) Point values: 7 points for each part of each problem, except 9 points for Problem A.

### Week 7 (5--9 Nov.)

Finish Chapter 3 (series).

Homework (due Monday 12 Nov.): Chapter 3: 11, 13, 14(a)--(d), 19. Point values: 10 points for each part of each problem. The syllabus has a hint for Problem 14(a).

### Week 8 (12--16 Nov.)

Start Chapter 4 (continuous functions).

Homework (due Monday 19 Nov.): Chapter 4: 1, 3, 4, 6, 7, 8, 9. Point values: 10 points for each part of each problem. In each of Problems 4 and 8, there are two statements to be proved; count each of these statements as a separate part. Consider Problem 7 to have two parts: one about f and one about g.

### Week 9 (19--21 Nov.) (short week, due to holidays)

Finish Chapter 4 (more on continuous functions).

Homework (due Wednesday 28 Nov.): Chapter 4: 10, 11, 12, 14, 15, 16, 18. Point values: 13 points per problem, except 22 points for Problem 11.

### Week 10 (26 Nov.--30 Nov.)

Start Chapter 5 (derivatives); review.

### Final exams week (3--7 Dec.)

• Tu 4 Dec., 8:00--10:00 pm: Review session 119 Fenton.
• Th 6 Dec., 10:15 am--12:15 pm: final exam.

## Publicly Available Documents Associated with this Course

Here is a list of publicly available documents associated with this course. The material is arranged in approximate chronological order: most recent items at the bottom. Links to written homework solutions, exams, and exam solutions will not work until after the corresponding written homework has been turned in or the corresponding exam has been given, and the links to sample exams and their solutions will not work until these items have been prepared.

Most files will be posted in pdf and AMSLaTeX versions.

## Important dates, according to the Academic Calendar at the registrar's office (not guaranteed!)

• Su 23 September: Last day to process a complete drop (100% refund, no W recorded).
• Sa 29 September: Last day to drop this course (100% refund, no W recorded; after this date, W's are recorded).
• Sa 29 September: Last day to process a complete drop (90% refund, no W recorded).
• Su 30 September: Last day to add this course.
• Su 30 September: Last day to withdraw from this course (100% refund, W recorded).
• W 3 October: Last day to change to or from audit.
• Su 7 October: Last day to withdraw from this course (75% refund, W recorded).
• Su 14 October: Last day to withdraw from this course (50% refund, W recorded).
• Su 21 October: Last day to withdraw from this course (25% refund, W recorded).
• Su 11 November: Last day to withdraw from this course (0% refund, W recorded).
• Su 11 November: Last day to change grading option or variable credits for this course. (Tuition penalties apply when reducing credits.)
• F 23 November: No classes. (Part of the Thanksgiving break.)

This page maintained by N. Christopher Phillips, email. Please email plain text (7 bit ASCII) only (no web page coded files, Microsoft Word documents, binary characters, etc.; see above for more).

Last significant change: 17 September 2018.