Krantz S.G. Transition to Analysis with Proof

CRC Press, 2018. — 362 p. — (Textbooks in mathematics). — ISBN: 978-1-1380-6414-0.Many of the classical texts in real analysis were written for a specialized audience of students planning to go on for an M.S. or Ph.D. in mathematics and then to pursue life as an academic. Yet, in our current situation, there are engineers and computer scientists and many others who need to understand the fundamentals of this subject. The purpose of this text is to address and teach the latter audience. We begin the book with a quick-and-dirty introduction to set theory and logic. It should be possible to cover that material in about three weeks. Then one can spend the rest of the semester studying real analysis. Of course, by real analysis we mean integration theory and differentiation theory and the theory of sequences and series of functions. That is the nexus of what this book is about. We eschew some of the beautiful but more difficult topics like the Stone–Weierstrass theorem, but we cover all the essential ideas at the heart of the matter. We certainly talk about the Weierstrass nowhere differentiable function and the Weierstrass approximation theorem and the Cantor set, but we do not belabor them. A student who wants to learn real analysis must do exercises. And we provide plenty of those. There is an exercise set at the end of each section. And the exercises are step-laddered. Both students and teachers can be confident that the first several exercises in each section are basic and accessible. Later exercises in each section are more challenging, or more open-ended. These are marked with an asterisk ∗ . Doing an exercise for a student is very much like working an example. And we exploit that connection for didactic purposes. It is my view that the way to learn real analysis is to analyze examples. So this book has plenty of worked examples. And plenty of figures to illustrate those examples. When practiced properly, real analysis is quite a visual subject. The book finishes with the Appendix: Elementary Number Systems, a Table of Notation, and a Glossary. Thus, the student will find that this text is a self-contained universe of real analysis. It will be a book that he/she will want to refer to in later years, and even as a professional. The index is quite detailed, making the topics of the book particularly accessible.