Math 331: Foundations of Analysis I

About Math 331: Goals

This course is an introduction to basic analysis of functions of a single variable. It presents a careful development of the real number system and the theory of calculus on the real line. This can be followed by further study in subsequent courses with extensions of the theory to R² and the complex plane. The course is designed to be a first encounter with rigorous, formal mathematics for serious mathematics students with a year of calculus.

Early mathematical education up through calculus focuses on learning how to carry out various mathematical operations. But the heart of mathematics does not lie in reproducing computational procedures or memorizing mathematical facts but in seeing why the facts are true, in grasping the arguments which show why the computation works, in understanding the connections which weave the individual pieces of mathematics into a theory. In Math 135, you were introduced to the proof techniques that are used to construct these arguments.

The discovery of this "heart" of mathematics begins by cultivating a deep curiosity about what underlies the methods and conclusions of mathematics. A desire to investigate how calculus is put together and what makes it tick is the primary prerequisite, along with mathematical maturity, for this course.

In developing this course with Professor Belding, we had three major goals in mind.

First, we wished to help students develop some understanding and control of the language of mathematics: definitions, theorems, and proofs.
Secondly, we wanted to present the central concepts and theory of calculus thoroughly and clearly enough to provide a secure foundation for more advanced mathematics courses and enable students to grasp some of the unity and beauty of the subject.
Finally, we wished to foster an appreciation for the living, human nature of mathematics, and a sense of how mathematics is created and evolves.

The transition to formal mathematics based on rigorous proofs is a necessary but difficult step. Reading and writing proofs requires a new level of precision and attention to detail. Rigorous definitions force us to rework our vague intuitive understanding of concepts such as limits or integrals.

This course will be an intellectual challenge and you should be proud of your achievement at the end of the term. You should feel that you could construct "the Calculus" starting from a small set of basic principles. You should feel ready to extend the concepts that you encounter to new situations and that your toolbox of mathematical proof techniques has been greatly enhanced. In short, you should feel like you are becoming a mathematician.

Text

Professor Belding and I wrote the text for this course with the students at Hobart and William Smith in mind. In this text we try to provide the demanding but supportive environment needed to make the transition to formal mathematics based on rigorous proofs successful.

We supply a context for the more difficult definitions and proofs by tracing the history behind the concepts. Hidden problems which motivate the design of a proof or wording of a definition are pointed out and discussed. The proofs have been crafted with the student in mind. Thus, many details are included which draw attention to points that may seem obvious or repetitive to more veteran mathematicians.

There are lots of problems! Nearly all problems require a proof of some kind. Some are simply intended to guide the review of material. These often require only slight modification of earlier arguments or immediate reflection on the meaning of a definition or theorem. However, written answers to such straightforward questions serve to reveal serious misconceptions early on. When an involved argument is required, the problem is frequently broken into steps to point toward fruitful questions and to reveal the structure and level of detail needed for a thorough proof.

This text does not pretend to be a comprehensive treatment of basic analysis. Instead, we have tried to present the main current of analysis faithfully and with enough spirit to inspire you to explore further.

Prerequisites and Expectations

This course assumes that you are going to be a math major or minor. Math 331 is one of the core courses for the mathematics major. Consequently, I expect that this course will be the one that you devote most of your attention to this term. Students usually need to spend 3 to 5 hours reading and working on problems for each lecture.

I assume that you have been very successful in Calculus I and II (and you still remember this material) and that you have had a good exposure to careful proofs through your work in Math 135 and Math 204. The skills that you learned in these two courses will be invaluable here. Also, when we construct the real numbers (yes, construct!), you will encounter some of the structures you worked with in linear algebra.

Labs

This course has an extra lab period. However, for the moment, I envision each class as a combination of lecture and lab or group work, along with some student presentations. (This course originally had two 2.5-hour labs each week!) These work periods are an excellent time for you to ask questions about the course material or get help with homework. You should carefully write up the solutions to the problems you work on so that you can use this material to study for exams.

Assessment

Homework, reading, and practice exercises will be assigned at the beginning of most class. I encourage working in small groups on practice problems. This can be very helpful in understanding the material.

About once a week, there will be a graded assignment consisting of selected problems to hand in. Unless otherwise stated, graded assignments are to be your own work without collaboration. These homework assignments should be done neatly. Work out the problems on scrap paper first and then make a final copy to hand in. Use pencil rather than pen to avoid having to cross out work. These problem sets when neatly and carefully done are an investment....you can use them to study for exams. No late assignments, please. I may also to give some announced, in-class quizzes on definitions or statements of theorems.

In addition to the aforementioned homework and quizzes, there will be three hour tests and a final exam. The dates are listed in the outline below. Tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.

Your course grade will be calculated as follows.

Homework and Quiz Grades: 22%
Attendance at two Departmental Seminars: 6%
Three 1-hour exams: 17% each
Final Exam: 21%

I also reserve the right to take class participation and attendance into account in determining final grades.

Attendance and Courtesy

Because of the nature of this course, its assignments, and its assessment, your attendance and participation is crucial. Mathematics is learned by regular, sustained, attentive effort over an extended period. Only when such effort has been invested will the concentrated study for an exam have any benefit. Therefore, attendance at class is required. Unexcused absences may adversely affect your grade. If you must miss a class for some reason beyond your control, talk to me about it in advance.

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.

Tips for Success

My best advice is to take good, complete notes during class. Even if you don't understand every detail during a lecture, with some patience you should be able to review each day's lecture and understand everything we did. If you don't, then you should come to see me as soon as possible. Here are a few simple things that you can do to be more successful in the course

Come to all classes.
Did I mention that you should take good, complete notes? If I think it is important enough to write down, so should you!
Do the readings and practice problems carefully.
Keep the answers to problems and notes on the readings in a journal.
Review the notes from the last class before coming to class again. In fact, I suggest recopying the notes! This makes studying for a test easy.
Come to see me for help whenever you need it, preferably before you get too far behind.
Start assignments early. Do not wait until just before an assignment is due to get help.
Ask questions in class. Ask questions about the homework and readings.

Outline of Weekly Readings

This assumes a fairly rapid pace through the listed materials. We may need to adjust this schedule based on our actual work in class.

Week	Topics and Readings
Weeks 1 and 2	Construction of the Real Numbers & their Properties. Chapter 1
Weeks 3 and 4	Limits: Chapter 2, Sections 1-3.
Monday, February 13	Exam 1
Weeks 5 and 6	Continuity. Chapter 2, Sections 4 and 5.
Weeks 7 and 8	Differentiation. Chapter 3 Sections 1-3.
Wednesday, March 8	Exam 2
Weeks 9, 10 and 11	Integration. Chapter 3, Sections 4-7
Week 12	Sequences. Chapter 4, Sections 1-2.
Monday, April 17	Exam 3
Week 13	Series. Chapter 4, Sections 3-4.
Week 14 and 15	Sequences and Series of Functions: Chapter 4, Sections 5 and 6.
Sunday, May 7	Final Exam at 3:00 PM