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How We Got From There To Here: A Story of Real Analysis
Robert Rogers, (Emeritus), Eugene Boman, (Emeritus)
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Front Matter
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Colophon
Acknowledgements
To the Instructor
I
In Which We Raise A Number Of Questions
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1
Prologue: Three Lessons Before We Begin
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1.1
Lesson One
1.2
Lesson Two
1.3
Lesson Three
2
Numbers, Real (
R
) and Rational (
Q
)
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2.1
Additional Problems
3
Calculus in the 17th and 18th Centuries
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3.1
Newton and Leibniz Get Started
Leibniz’s Calculus Rules
Leibniz’s Approach to the Product Rule
Newton’s Approach to the Product Rule
3.2
Power Series as Infinite Polynomials
3.3
Additional Problems
4
Questions Concerning Power Series
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4.1
Taylor’s Formula
4.2
Series Anomalies
4.3
Additional Problems
II
Interregnum
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5
Joseph Fourier: The Man Who Broke Calculus
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5.1
Joseph Fourier and His Series
III
In Which We Find (Some) Answers
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6
Convergence of Sequences and Series
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6.1
Sequences of Real Numbers
6.2
The Limit as a Primary Tool
6.3
Divergence
6.4
Additional Problems
7
A “Tayl” of Three Remainders
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7.1
The Integral Form of the Remainder
7.2
Lagrange’s Form of the Remainder
7.3
Cauchy’s Form of the Remainder
7.4
Additional Problems
8
Continuity: What It Isn’t and What It Is
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8.1
An Analytic Definition of Continuity
8.2
Sequences and Continuity
8.3
The Definition of the Limit of a Function
8.4
The Derivative, An Afterthought
8.5
Additional Problems
9
Intermediate and Extreme Values
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9.1
Completeness of the Real Number System
9.2
Proof of the Intermediate Value Theorem
9.3
The Bolzano-Weierstrass Theorem
9.4
The Supremum and the Extreme Value Theorem
9.5
Additional Problems
10
Back to Power Series
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10.1
Uniform Convergence
10.2
Uniform Convergence: Integrals and Derivatives
Cauchy Sequences
10.3
Radius of Convergence of a Power Series
10.4
Boundary Issues and Abel’s Theorem
11
Back to the Real Numbers
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11.1
Infinite Sets
11.2
Cantor’s Theorem and Its Consequences
12
Epilogues
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12.1
On the Nature of Numbers: A Dialogue (with Apologies to Galileo)
Additional Problems
12.2
Building the Real Numbers
The Decimal Expansion
Cauchy Sequences
Dedekind Cuts
Back Matter
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Bibliography
Index
Chapter
10
Back to Power Series
10.1
Uniform Convergence
10.2
Uniform Convergence: Integrals and Derivatives
10.3
Radius of Convergence of a Power Series
10.4
Boundary Issues and Abel’s Theorem
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