This is a no-frills introduction to real analysis that is suitable for a basic one-semester undergraduate course. It is designed to serve both future mathematicians as well as students not intending to pursue mathematics in graduate school. For this reason it is perhaps a better match for classes with a mixture of abilities, motivations and career plans than commonly available alternative texts. The book is freely available here, and an inexpensive hardcopy is also available from Lulu.

The contents are quite standard: real numbers, sequences and series, continuous functions, the derivative, the Riemann integral, and sequences of functions. The last chapter is an introduction to metric spaces. This book is similar in many respects to Bartle and Sherbert’s *Introduction to Real Analysis* (and indeed the author provides a table of correspondences between sections of the two books).

The differences between the two books are illuminating. Lebl seems to be particularly sensitive to the variety of students who might use his book. He is aware that some of them will be future high school teachers of calculus. This awareness is reflected in his pedagogical approach. For example, Lebl chooses to follow the Darboux approach to the Riemann integral. It is perhaps less elegant than Bartle and Sherbert’s approach, but less sophisticated and more intuitive. Lebl also avoids proofs by contradiction as much as possible. He does this not so much for philosophical reasons as because he thinks that it is a common source of trouble for beginning students. It is important to note that Lebl is not dumbing-down real analysis: his approach is clean, clear and rigorous. He’s just not tempted to get too fancy.

One thing I particularly like about his approach is the way that Lebl organizes the book with a kind of capstone theorem. He uses Picard’s theorem on existence and uniqueness of solutions of ordinary differential equations to pull together many of the things students have learned to provide a direct, more-or-less constructive proof. Then in the chapter on metric spaces he offers another proof using the contraction mapping theorem.

One thing I missed in the book are more counterexamples. There is a little too much emphasis on rigorous ratification of calculus. Why should students find it valuable to work hard to learn to prove things that no one could possibly doubt?

This is an attractive book, one well worth considering for anyone about to teach introductory real analysis. One of Lebl’s reasons for making the book freely available is to allow others to customize it to suit their purposes. It also allows for relatively quick correction of errors, especially compared to the usual long wait for the next publication cycle.

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.