Milne Open Textbooks

**Author(s):**
Eugene Boman and Robert Rogers

**Publication Date:** February 18, 2014

**ISBN:** 978-1-956862-03-4

**OCLC:** 903663475

**Affiliation: **SUNY Fredonia

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.

**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

- 2.1 Newton and Leibniz Get Started
- 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

- 4 Convergence of Sequences and Series
**Epilogue**- Epilogue: On the Nature of Numbers
- Epilogue: Building the Real Numbers
- The Decimal Expansion
- Cauchy sequences
- Dedekind Cuts