Transition to Analysis with Proof

Steven G. Krantz is a professor at Washington University in St. Louis where he teaches mathematics. He received his Ph.D. from Princeton University and since then has taught at UCLA, Princeton University, and Pennsylvania State University. Dr. Krantz has written over 175 scholarly papers and more than 65 books. He is the founding editor of the Journal of Geometric Analysis. He was named a fellow of the American Mathematical Society and has received the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize.

This book is designed for undergraduate students and provides an introduction to analysis including an introduction to proofs. Namely, it starts with basic logic along with set theory and suggests plenty of instructive examples on how to do a correct mathematical reasoning. Then, he moves to functions, sequences, basic topology, differentiation, and integration. The text ends with power series and elementary transcendental functions. Each section of the book has a clear and convenient structure. It opens with "Preliminary remarks", which in a few sentences explain what the following passage is about. Then, the main text follows in which every statement is equipped with an instructive proof. Examples and exercises help master the material of the topic. \A look back" blocks consist of a few simple questions (like "What is a monotone sequence?") and serve as a test if you understand the basics.General remarks. The language of the book is quite vivid and lively, at the same time it is available for non-native students. The text has useful navigation tools like glossary, index, and the table of notations. The only disadvantage, in my opinion, is the lack of solutions and answers to the exercises.Nikita Evseev (Novosibirsk)

• Chapter 1 Basic Logic; Chapter 2 Methods of Proof; Chapter 3 Set TheoryChapter 4 Relations and Functions; Chapter 5 Essential Number Systems Chapter 6 Sequences; Chapter 7 Series of Numbers; Chapter 8 Basic Topology Open and Closed Sets; Chapter 9 Limits and Continuity of Functions; Chapter 10 Differentiation of Functions; Chapter 11 The Integral; APPENDIX: Construction of The Weierstrass Nowhere Differentiable Function; Chapter 12 Sequences and Series of Functions; APPENDIX: Proof of the Weierstrass Approximation Theorem; Chapter 13 Elementary Transcendental Functions APPENDIX I Elementary Number Systems Table NotationGlossary Bibliography, Index