How We Got from There to Here: A Story of Real Analysis
Edition: First edition
Publisher: Milne Publishing
Published: June 14, 2014
Place of publication: Geneseo, NY
Format: Paperback Open Textbook
Number of pages: 220
ISBN-13: 9781312348691
Copyright date: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License
First sentence: The irony of this section is that it exists to tell you that this book was not written for you; it was written for your students.
Description: This is an Open Textbook. Online and PDF available at milneopentextbooks.org
Notes: This is an Open Textbook.
Table of contents: title: I: In Which We Raise a Number of Questions; title: 1 Numbers, Real (R) and Rational (Q); title: 2 Calculus in the 17th and 18th Centuries; title: 2.1 Newton and Leibniz Get Started; title: 2.1.1 Leibniz’s Calculus Rules; title: 2.1.2 Leibniz’s Approach to the Product Rule; title: 2.1.3 Newton’s Approach to the Product Rule; title: 2.2 Power Series as Infinite Polynomials; title: 3 Questions Concerning Power Series; title: 3.1 Taylor’s Formula; title: 3.2 Series Anomalies; title: II Interregnum; title: Joseph Fourier: The Man Who Broke Calculus; title: III In Which We Find (Some) Answers; title: 4 Convergence of Sequences and Series; title: 4.1 Sequences of Real Numbers; title: 4.2 The Limit as a Primary Tool; title: 4.3 Divergence; title: 5 Convergence of the Taylor Series: A “Tayl” of Three Remainders; title: 5.1 The Integral Form of the Remainder; title: 5.2 Lagrange’s Form of the Remainder; title: 5.3 Cauchy’s Form of the Remainder; title: 6 Continuity: What It Isn’t and What It Is; title: 6.1 An Analytic Definition of Continuity; title: 6.2 Sequences and Continuity; title: 6.3 The Definition of the Limit of a Function; title: 6.4 The Derivative, An Afterthought; title: 7 Intermediate and Extreme Values; title: 7.1 Completeness of the Real Number System; title: 7.2 Proof of the Intermediate Value Theorem; title: 7.3 The Bolzano-Weierstrass Theorem; title: 7.4 The Supremum and the Extreme Value Theorem; title: 8 Back to Power Series; title: 8.1 Uniform Convergence; title: 8.2 Uniform Convergence: Integrals and Derivatives; title: 8.2.1 Cauchy Sequences; title: 8.3 Radius of Convergence of a Power Series; title: 8.4 Boundary Issues and Abel’s Theorem; title: 9 Back to the Real Numbers; title: 9.1 Infinite Sets; title: 9.2 Cantor’s Theorem and Its Consequences; title: Epilogue; title: Epilogue: On the Nature of Numbers; title: Epilogue: Building the Real Numbers; title: The Decimal Expansion; title: Cauchy sequences; title: Dedekind Cuts