# How We Got from There to Here: A Story of Real Analysis

Robert Rogers, State University of New York

Eugene Boman, The Pennsylvania State University

Copyright Year: 2014

ISBN 13: 9781312348691

Publisher: Open SUNY

Language: English

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## Conditions of Use

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CC BY-NC-SA

## Reviews

This book covers all standard content of an introductory Real Analysis course. read more

This book covers all standard content of an introductory Real Analysis course.

The book's content is perfectly accurate, I could not detect any error.

This book presents Real Analysis in a way that seems to the reviewer absolutely new and quite interesting and exciting. Hopefully it will also encourage the release of more books with a similar approach.

The book's prose is easy to follow and it avoids systematically, on purpose, unnecessarily technical terms in order to increase readability. In fact, the whole point of this particular book is to make as easy and effortless as possible to undergraduate students the task of understanding the most central concepts of real analysis, that unfortunately look, at first sight, unnatural and hard to grasp.

The book presents the topics of Real Analysis in the order in which they were historically developed. The consistency of the book comes automatically out of this.

The text is finely divided in many sections and therefore highly modular.

Once again, topics in this text are sorted in historical order. On one side, this is not the most logical order. On the other side, learning how and why new concepts had to be introduced during the development of the theory does help the reader to understand the reason for the introduction of concepts and definitions otherwise quite hard to grasp.

This book is based on the pretext package, a very solid package to produce books in the most widespread online (e.g. HTML) and offline (e.g. PDF) formats. In particular, its HTML interface is excellent.

I could not find any grammar problem (not, though, that English is not my native language).

This category does not really apply to math books in general, and to this in particular.

This undergraduate level text is, in my experience, a precious and unique Real Analysis text because tries to solve, by a historical approach, one of the main problem of teaching this topic, namely the fact that the most important concepts -- in particular the 'epsilon-delta' definition of limits -- are quite unintuitive and mastering them is a challenge for most students (it certainly was for me when I first met it!). Moreover the book does an excellent job at pushing strongly the reader from the very beginning into being an active reader, for instance by insisting that he/she should always verify and complete the details of calculations in the book while reading the book -- always keep pencil and scrap paper while reading a math book! I agree with the authors that this is the best -- and perhaps only -- way to really learn from a math book. The whole book is architectured so to push the reader to "enter" the book and "fill the dots" by him/herself while learning the material. Of course hints are provided for the most challenging points.

Perhaps one can say that the keyword of this book is 'motivations'. Unfortunately often in math publications the reader is left clueless of which motivations lead to definitions that seem to appear 'out of the blue', causing sometimes a complete loss of interest in the subject and anyways making unnecessarily hard learning the material. This book shows that a historical approach provides an automatic solution to this problem. It would be very interesting to have a similar text about Complex Analysis.

Besides standard material for an analysis book, this text runs on themes motivated by discussions of the history of analysis. Along with that come some topics that would not normally appear in an analysis text, but that broaden the appeal of the... read more

Besides standard material for an analysis book, this text runs on themes motivated by discussions of the history of analysis. Along with that come some topics that would not normally appear in an analysis text, but that broaden the appeal of the subject. These include Fourier series and their role in the development of analysis, Snell's law (how Leibnitz derived it, and later in the text how Bernoulli used it to solve the Brachistochrone problem), some number theory (Fermat's little theorem), and how power series can be used to solve differential equations.

The text is accurate. Many of the proofs appear only as sketches that the reader will hopefully fill in accurately. The authors encourage readers to talk to others, hopefully making it likely that mistakes in readers' proofs will be caught.

This is classical material, arranged to a large extent according to historical discovery.

At a first reading, I enjoy the informal style and reminders that the reader needs to work out the details of the calculations. I imagine that this might get old for some of those learning for the first time and reading it multiple times. The figures could be better made and (this is the reason for a ranking of 4 rather than 5) axes are not always labeled.

The text is consistent in both terminology and style.

It is possible to assign small reading sections. The text is designed to initiate ideas early (such as in the opening discussion of the meaning of 'number") and come back to those ideas later.

The text opens with three 'lessons' that i) get the reader thinking about numbers and definitions, ii) encourage the reader to work out the details of computations and algebraic manipulations given in the text, and iii) give some philosophical insights into solving problems. After that, rather than just giving the reader results of analysis, Part I raises questions on numbers (building on the opening lesson) and series. Only after another interlude on Fourier series does the text focus more on finding answers to the problems. The construction of real numbers is left to the end, at which point the student has more appreciation for thinking about what numbers are. An aspect of the book that I particularly appreciate is how much thought the authors have given to initiating ideas that they come back to later--such as the theme of understading what a number is, which opens the first lesson of the text and also rounds out the text in the concluding material.

Besides a normal index. there are lists of Theorems, Lemmas, Corollaries, Definitions and Problems at the end of the book, all with links to the pages in the book. The index might lack complete comprehesiveness (I didn't find 'Snell's Law' there), but that does not really matter since one can search in a .pdf file without an index.

The grammar is great.

It is rare for a math text to include cultural/historic background as a truly important part of the presentation of the material.

The text, being very enjoyable to work through and leaving it the reader to complete proofs, would seem to work very well with (but not only with) a flipped classroom style of teaching. Something that is usually underdeveloped in mathematics education is letting students discover interesting problems; this text makes some progress towards doing that by telling the tale of how interesting problems in analysis were come upon.

From Number to Cantor's Theorem, this book brings you on a journey of the development of mathematical analysis. Several important stops along the way include Taylor Series, the Bolazano-Weierstrass Theorem, and Cauchy Sequences, I cannot think of... read more

From Number to Cantor's Theorem, this book brings you on a journey of the development of mathematical analysis. Several important stops along the way include Taylor Series, the Bolazano-Weierstrass Theorem, and Cauchy Sequences, I cannot think of any notable omissions along the road.

Spot on, though the text is built on problems that leave a lot of work for the reader.

The approach the authors take is essentially timeless, in that it brings us to modern analysis. I imagine fifty years from now we could still look at this book as a very good exposition of how we got to where we are in twentieth century mathematics, and that will still be quite relevant for our moderately advanced undergraduates and casual mathematically curious students.

There are times when the prosaic nature of the narrative is a little strained, but the intention is to make the text more accessible. It is not a serious detraction, nor does it significantly get in the way of the text's movement. In the final analysis, it is a pleasure to read and the text moves logically.

Very strong. The authors have a genuine interest in the story they weave and keep the tale together throughout the text.

The text builds naturally through the history of mathematical analysis, so modularity is not itself a strong objective. With that said, you can reasonable go to any section and learn from it, but this text intentionally and appropriately is intended to be taken more as a sequential whole than in unrelated parts.

Time is a natural progression -- the time of mathematics development and how results build from one generation of mathematicians to the next -- and this text flows naturally through this development.

The text is well put together and easy to use. The links in the PDF are useful.

I observed no significant grammatical issues

This is exactly the approach to deliver cultural literacy to the budding mathematically inquisitive student. This text contains much of the lore of mathematical analysis that is appropriate for our undergraduate students.

This is a delightful text, and exactly how we should develop undergraduate analysis to our mathematical studies students. There are several classic results, but more importantly it shows how analysis develops, and hence why modern analysis has the character it does. This is how we should prepare our students for advanced undergraduate or graduate analysis, giving them a perspective of mathematics. For example, instead of taking an informal aside to explain the countability of the rationals, this is done naturally through the text (Problem 201).

The text leaves many details, including most proofs, as exercises. If you adopt this text you will probably want to take many of these as illustrative examples in your lecture development. Hence, there would be a significant amount of preparatory work to deliver this material, but it gives a very good framework to base your class on.

## Table of Contents

- 1 Numbers, Real (R) and Rational (Q)
- 2 Calculus in the 17th and 18th Centuries
- 3 Questions Concerning Power Series
- 4 Convergence of Sequences and Series
- 5 Convergence of the Taylor Series: A “Tayl” of Three Remainders
- 6 Continuity: What It Isn't and What It Is
- 7 Intermediate and Extreme Values
- 8 Back to Power Series
- 9 Back to the Real Numbers

## Ancillary Material

Submit ancillary resource## About the Book

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context. However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context.

This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.

## About the Contributors

### Authors

**Robert Rogers** received his BS in Mathematics with Certification in Secondary Education from Buffalo State College in 1979. He earned his MS in Mathematics from Syracuse University in 1980 and his Ph.D. in Mathematics from the University of Buffalo in 1987, specializing in Functional Analysis/Operator Theory. He has been on the faculty of the State University of New York at Fredonia since 1987 where he is currently Professor of Mathematics. He is a recipient of the SUNY Fredonia President’s Award for Excellence in Teaching and the MAA Seaway Section’s Clarence F. Stephens’ Award for Distinguished Teaching. He is also a recipient of the MAA Seaway Section’s Distinguished Service Award. He is currently the editor of the New York State Mathematics Teachers’ Journal.

**Eugene Boman** received his BA from Reed College in 1984, his MA in 1986 and his Ph.D. in 1993, both of the latter were from the University of Connecticut. He has been teaching math at The Pennsylvania State University since 1996, first at the DuBois campus (1996-2006) and then at the Harrisburg campus. In 2008 he won the Carl B. Allendorfer Award for excellence in expository mathematical writing from the editors of Mathematics Magazine, for the article “Mom! There’s an Astroid in My Closet” (Mathematics Magazine, Vol. 80 (2007), pp. 247-273).