Understanding Real Analysis

This is a textbook designed to teach students who are new to analysis what it’s all about. It takes a very gentle approach and seems to be tuned especially for students of modest ability or limited mathematical maturity. These students are often left behind by treatments that move too quickly, attempt to treat a lot of topics, or assume a more sophisticated audience. The contents include about a semester’s worth of material.

The path Zorn takes is based on several very reasonable principles. These include: building on calculus basics; focusing on mathematical proof, structure and language; staying with the basics; offering many examples and many solved exercises; and gradually increasing technical sophistication. Zorn does not assume that students have previously taken an “introduction to proof” course. Instead, the first chapter gently introduces the basic ideas of proof and proof-writing along with a discussion of functions, sets and real numbers. Zorn is serious about teaching careful use of mathematical language. He is clearly aware of the difficulties many students have — learning to write mathematical arguments coherently, mastering the vocabulary, and learning the uses of, and distinguishing between, logical quantifiers.

After the introductory chapter, the author treats sequences and series of real numbers, limits and continuity, derivatives, and integrals in successive chapters. Zorn takes pains, especially in the early chapters, to make his exposition as clear and explicit as possible, sparing no details. So, for example, he defines the concept of limit, and slowly unpacks it, element by element, for the student. Part of the “stay with the basics” program means that there is no discussion of compactness, no mention of series of functions, and only a very basic treatment of the Riemann integral.

There are plenty of exercises. They tend to follow a pattern where an exercise that is not completely straightforward is broken into multiple parts to guide the student to a solution. Fairly complete solutions to odd-numbered exercises are provided in an appendix.

This is obviously a book markedly different from Rudin’s Principles of Mathematical Analysis or any of its more recent analogs. Zorn has a student audience in mind that needs, or would benefit from, a much gentler approach. Stronger students would likely find the pace too slow. Supplementary material from another text could help with this, especially if the instructor is facing a class with a broad range of talent, interest, and maturity.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.