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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.
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.
"This book is well written and so it is also very convenient as a textbook for a standard one-semester course in real analysis." (Petr Gurka, zbMATH 1454.26001, 2021)
"This is a well-written book with definitions embedded in the text-these are easily identified by bold type throughout the work. The theorems and proofs are set apart from the text and appear in boxes that follow discussions that motivate them. ... Summing Up: Recommended. Lower- and upper-division undergraduates; researchers and faculty." (J. R. Burke, Choice, Vol. 54 (7), March, 2017)
"Its objective is to make the reader understand the thought processes behind the proofs. In this it succeeds admirable, and then book should be in every mathematical library, public and private. ... The book is excellently produced with many coloured diagrams." (P. S. Bullen, Mathematical Reviews, January, 2017)
"I think this is indeed a fabulous book for the kind of course I just suggested. I think that it will indeed serve as Kane projects it should, and the surviving student will truly know a good deal about writing a mathematical proof, in fact, about thinking about the problems and assertions beforehand and then going about the task of constructing the proof." (Michael Berg, MAA Reviews, August, 2016)
"This is a well-written book with definitions embedded in the text-these are easily identified by bold type throughout the work. The theorems and proofs are set apart from the text and appear in boxes that follow discussions that motivate them. ... Summing Up: Recommended. Lower- and upper-division undergraduates; researchers and faculty." (J. R. Burke, Choice, Vol. 54 (7), March, 2017)
"Its objective is to make the reader understand the thought processes behind the proofs. In this it succeeds admirable, and then book should be in every mathematical library, public and private. ... The book is excellently produced with many coloured diagrams." (P. S. Bullen, Mathematical Reviews, January, 2017)
"I think this is indeed a fabulous book for the kind of course I just suggested. I think that it will indeed serve as Kane projects it should, and the surviving student will truly know a good deal about writing a mathematical proof, in fact, about thinking about the problems and assertions beforehand and then going about the task of constructing the proof." (Michael Berg, MAA Reviews, August, 2016)