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How We Got From There To Here: A Story of Real Analysis
Robert Rogers, Eugene C. Boman
Contents
Index
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Contents
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Front Matter
Colophon
Acknowledgements
1
To the Instructor
2
Prologue: Three Lessons Before We Begin
Lesson One
Lesson Two
Lesson Three
I
In Which We Raise A Number Of Questions
3
Numbers, Real (
\(\RR\)
) and Rational (
\(\QQ\)
)
Additional Problems
4
Calculus in the 17th and 18th Centuries
Newton and Leibniz Get Started
Power Series as Infinite Polynomials
Additional Problems
5
Questions Concerning Power Series
Taylor's Formula
Series Anomalies
Additional Problems
II
Interregnum
6
Joseph Fourier: The Man Who Broke Calculus
Joseph Fourier and His Series
III
In Which We Find (Some) Answers
7
Convergence of Sequences and Series
Sequences of Real Numbers
The Limit as a Primary Tool
Divergence
Additional Problems
8
A “Tayl” of Three Remainders
The Integral Form of the Remainder
Lagrange's Form of the Remainder
Cauchy's Form of the Remainder
Additional Problems
9
Continuity: What It Isn't and What It Is
An Analytic Definition of Continuity
Sequences and Continuity
The Definition of the Limit of a Function
The Derivative, An Afterthought
Additional Problems
10
Intermediate and Extreme Values
Completeness of the Real Number System
Proof of the Intermediate Value Theorem
The Bolzano-Weierstrass Theorem
The Supremum and the Extreme Value Theorem
Additional Problems
11
Back to Power Series
Uniform Convergence
Uniform Convergence: Integrals and Derivatives
Radius of Convergence of a Power Series
Boundary Issues and Abel's Theorem
12
Back to the Real Numbers
Infinite Sets
Cantor's Theorem and Its Consequences
13
Epilogues
On the Nature of Numbers: A Dialogue (with Apologies to Galileo)
Building the Real Numbers
Back Matter
Bibliography
Index
Authored in PreTeXt
How We Got From There To Here: A Story of Real Analysis
Robert Rogers
Department of Mathematics and Computer Science
SUNY, Fredonia
Fredonia, New York, USA
Robert.Rogers@fredonia.edu
Eugene C. Boman
Department of Mathematics and Computer Science
Penn State, Harrisburg campus
Harrisburg, PA, USA
ecb5@psu.edu
Colophon
Acknowledgements