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

(3 reviews)

Robert Rogers, State University of New York

Eugene Boman, The Pennsylvania State University

ISBN 13: 9781312348691

Publisher: Open SUNY

Language: English

CC BY-NC-SA

## Reviews

Reviewed by Roberto De Leo, Associate Professor, Howard University on 2/24/21

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

Reviewed by Patrick Shipman, Associate Professor, Colorado State University on 1/7/16

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

Reviewed by Clark Gaylord, Adjunct, Virginia Tech on 6/10/15

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

• 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

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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.