This is a textbook on proof writing in the area of analysis, balancing a survey of the core concepts of mathematical proof with a tight, rigorous examination of the specific tools needed for an understanding of analysis. Instead of the standard "transition" approach to teaching proofs, wherein students are taught fundamentals of logic, given some common proof strategies such as mathematical induction, and presented with a series of well-written proofs to mimic, this textbook teaches what a student needs to be thinking about when trying to construct a proof. Covering the fundamentals of analysis sufficient for a typical beginning Real Analysis course, it never loses sight of the fact that its primary focus is about proof writing skills. This book aims to give the student precise training in the writing of proofs by explaining exactly what elements make up a correct proof, how one goes about constructing an acceptable proof, and, by learning to recognize a correct proof, how to avoid writing incorrect proofs. To this end, all proofs presented in this text are preceded by detailed explanations describing the thought process one goes through when constructing the proof. Over 150 example proofs, templates, and axioms are presented alongside full-color diagrams to elucidate the topics at hand.
About the Author
Jonathan Michael Kane is an emeritus professor of Mathematical and Computer Sciences at the University of Wisconsin - Whitewater and an honorary fellow of the Department of Mathematics at the University of Wisconsin - Madison. He has published papers in several complex variables, probability, algorithms, and the relationship between gender and culture in mathematics performance. He has taught dozens of courses in mathematics, statistics, actuarial mathematics, and computer science. Dr. Kane plays a major role in contest mathematics by chairing the American Invitational Mathematics Exam Committee, cofounding and coordinating the annual online Purple Comet! Math Meet, and teaching at the AwesomeMath summer program.