Milne Open Textbooks

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

Author(s): and

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.

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



  • Peer-review: “Problems stimulate students to independent thinking in discovering analysis. The presentation is engaging and motivates the student with numerous examples, remarks, illustrations, and exercises. Clearly, it is a carefully written book with a thoughtful perspective for students.”
    –Osman Yurekli, Ph.D.
  • Review by:  The Mathematical Association of America.
  • Review by: Open Textbook Library.


Using this text? We'd love to know more about you: fill out this short questionnaire to let us know!

Prologue: Three Lessons Before We Begin

I: In Which We Raise a Number of Questions

1 Numbers, Real (R) and Rational (Q)

2 Calculus in the 17th and 18th Centuries

2.1 Newton and Leibniz Get Started
2.1.1 Leibniz’s Calculus Rules
2.1.2 Leibniz’s Approach to the Product Rule
2.1.3 Newton’s Approach to the Product Rule
2.2 Power Series as Infinite Polynomials

3 Questions Concerning Power Series

3.1 Taylor’s Formula
3.2 Series Anomalies

II Interregnum

Joseph Fourier: The Man Who Broke Calculus

III In Which We Find (Some) Answers

4 Convergence of Sequences and Series

4.1 Sequences of Real Numbers
4.2 The Limit as a Primary Tool
4.3 Divergence

5 Convergence of the Taylor Series: A “Tayl” of Three Remainders

5.1 The Integral Form of the Remainder
5.2 Lagrange’s Form of the Remainder
5.3 Cauchy’s Form of the Remainder

6 Continuity: What It Isn’t and What It Is

6.1 An Analytic Definition of Continuity
6.2 Sequences and Continuity
6.3 The Definition of the Limit of a Function
6.4 The Derivative, An Afterthought

7 Intermediate and Extreme Values

7.1 Completeness of the Real Number System
7.2 Proof of the Intermediate Value Theorem
7.3 The Bolzano-Weierstrass Theorem
7.4 The Supremum and the Extreme Value Theorem

8 Back to Power Series

8.1 Uniform Convergence
8.2 Uniform Convergence: Integrals and Derivatives
8.2.1 Cauchy Sequences
8.3 Radius of Convergence of a Power Series
8.4 Boundary Issues and Abel’s Theorem

9 Back to the Real Numbers

9.1 Infinite Sets
9.2 Cantor’s Theorem and Its Consequences


Epilogue: On the Nature of Numbers

Epilogue: Building the Real Numbers

The Decimal Expansion
Cauchy sequences
Dedekind Cuts

Eugene Boman

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 Asteroid in My Closet” (Mathematics Magazine, Vol. 80 (2007), pp. 247-273).

Robert Rogers

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.