Writing Proofs in Analysis by Jonathan M. Kane (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Best introductory book to non-measure-theoretic mathematical analysis and proof skills! This is the book for beginner to read alongside with Rudin, or before Rudin (PMA).

⭐This book was a really great idea for math lovers everywhere. It allows us to relive the bygone days of introductory real analysis. Reading it could give beginning students in analysis a leg up in their studies!

⭐First, I want to note the near date posting of this review after the first reviewer. This is simply a coincidence. I have not completed the book but I am excited enough by its approach to post a “heads up” review. I think it is sensational. Kane is spelling out the steps involved in proving a concept (theorem) in math and then clearly explaining why it is demonstrably the case; step-by-step; idea-by-idea; concept-by-concept. And in this case, template-by-template. Who knew? I write with mild facetiousness given my frustration with other texts that claim to be “teaching” how to do mathematical proofs. It has been my experience that pedagogy has been sorely lacking in this endeavor, Well, if you have shared this experience, then help is on the way.Thank you, Professor Kane!P.S. It’s kinda of fun, too, in a weird sort of way.

⭐This is really a wonderful book, one of a kind! The title says it’s about writing proofs, and the emphasis in the book is on that, but it gives much more: it presents in a best possible way the foundations of Math Analysis for beginners in the subject. I, like the author of the book himself, have taught intro to advanced math courses for many years. In such courses, students are supposed to learn how to argue when discussing math and how to properly write their arguments down. Elements of Math Analysis is one topic covered in these courses. I know first hand that some of the students taking this course suffer not understanding what Math Analysis is about which blocks their ability to think about making arguments on the subject. These students are unable to find/organize arguments when proving a claim. There are others who are pass all that, but have troubles writing down their arguments (they either skip steps in it, or are unnecessarily wordy). This book is a perfect source to learn from for the type of students mentioned here: it teaches how to think of/understand Math Analysis, and how to professionally lay out the arguments made. This is achieved in a different from the usual approach way: the concepts from Math and Math Logic (methods of proof etc.) used in the book are introduced when and where they are needed and in a rigorous fashion with very well explanations; all claims/theorems are clearly formulated; the arguments built are motivated in appropriate discussions the outcome of which is in the form of concise, elegant, and to the point (not a word in them is unnecessary) written down proofs. This is done meticulously for ALL claims/theorems in the book! This is not all! The relationship between different claims from the point of view of developing the subject is explained which helps the students understand what Math Analysis is about. Very often, as a result of the discussions in pursuing arguments for a proof, the book comes out with more than one proof of a claim. This I think will convince the students that proofs are not to be memorized, but well understood and applied with creativity. Very importantly, something that makes the book stand out – the structure of the body of a proof, how a proof has to be organized, is thoroughly explained so that the students would know how to order their thoughts in writing down a proof. It deserves a mention that the masterful presentation in the book allows quite a lot of math and discussion thereof to be covered on a bit more than 300 pages! Yes, the title of the book says it’s about writing proofs, and it is emphatically about that, but the book gives much, much more! This is truly a wonderful book!

⭐Clear, and excellent

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