Math 247: Elementary Real Analysis,
Fall Semester 1997

Course Description

A careful and rigorous examination of the elementary theory of calculus through the study of properties of the real line.

 

Prerequisites

Calculus III and an introductory course in linear algebra. The purpose of the linear algebra course is to assure some experience with mathematical proofs.

 

Text:  Foundations of Analysis   by D. F. Belding and Kevin J. Mitchell, Prentice Hall,             1991.

 

Unfortunately, this book is out of print. The local bookstore was able to find an adequate number of copies for my class, but I cannot rely on this for future classes. The text is not ideal for my purposes, but it is the best that I have found.

 

My aims for the course this semester.

My goals are to do a careful and rigorous but simple "stripped down" introduction to the real analysis of the line with an emphasis on the writing of proofs. Whenever possible I avoid the more abstract concepts such as compactness and connectedness . This occasionally requires me to prepare handouts with alternative proofs. For example, the text incorporates the modern notion of compactness (although the authors avoid the actual term) to prove that a continuous function defined on a closed interval is uniformly continuous. I will devise a proof using the Bolzano-Weierstrass theorem.

 

My method of teaching this course.

I use the "lecture" method. I think it is important for students at this stage to see proofs as well as explanations of how one can discover and write proofs in analysis. I use a "Brechtian" approach. For example, in proving properties of limits I talk about the problem of finding a proof, show how it is possible to "think backwards" in order to discover a rigorous proof, and then give the rigorous proof based on my explorations. I emphasize the importance of distinguishing between the preliminary thinking in order to discover a proof, and the actual proof. I assign a lot of homework in which students are required to give rigorous proofs. One of the linguistic problems that students in analysis must face is the appropriate use of quantifiers. This adds a dimension to the difficulties of writing. To a great extent I think of myself as a writing teacher as well as a mathematics teacher. In fact, I believe, based on many conversations with students over many years, that students who do well in English composition classes are more likely to succeed here.

 

There is some movement in the department to replace this course, which is required for math majors, with a computer based modeling course. I don't think this is a good idea because I think the students do need more experience with rigorous mathematics with some depth. They also need the writing experiences that normally accompany this course.