Introduction to Analysis

In the preface, the author says that this book “is designed to bridge the gap between the intuitive calculus normally offered at the undergraduate level and the sophisticated analysis courses the student encounters at the senior or first-year-graduate level”. He accomplishes that goal handily, in a quite workmanlike manner. The book under review is in its fifth edition, and presumably well-tested in the classroom. However, the statement quoted above (one, I should note, I have seen several times in comparable textbooks) raises questions that I’ll address below.

The collection of topics and their sequence should be familiar to anyone who has ever picked up a text on undergraduate analysis. Beginning with preliminaries on basic set theory, the author takes up sequences, limits of functions, continuity, differentiation, the Riemann integral, infinite series, and then sequences and series of functions. Perhaps the only variation from the usual path is to postpone the treatment of numerical series until near the end. The treatment is direct and readable, with clearly written proofs and an awareness of the areas where students are likely to have difficulties. If you were told only two days in advance that you were teaching an undergraduate analysis course, this text would work for you. There are no surprises and the collection of exercises and projects (multi-step problems with direction and assistance) have been carefully selected.

However, I do have two concerns about the book. One of them pertains directly to the bridge-the-gap comment quoted above. The undergraduate analysis course is taken by many mathematics majors (and is often required), yet a majority of them do not go to graduate school or even take another analysis course. Further, this course is regarded by many as the most difficult course in the major. These students need a sense that they are doing more than taking a course intermediate to something else that will never follow. They also need some better motivation.

My second concern is that the book seems a little flat. Excitement for the subject just isn’t there. I think that a reader could easily get the sense that the book is primarily a ratification of the methods and calculations of calculus. Yet one opportunity the first course in analysis offers, by judicious choice of examples, is to give students some doubts, so they begin to think that the theorems aren’t so obvious and there really is a point to the rigor. Although this first course is not the place to investigate a multitude of pathological examples, it would be useful to see the Cantor set, for example. The real numbers are strange and mysterious. Students should never get the sense that real analysis is just another requirement.

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.