# Understanding Analysis

###### Stephen D. Abbott
Publisher:
Springer Verlag
Publication Date:
2001
Number of Pages:
257
Format:
Hardcover
Series:
Price:
39.95
ISBN:
0-387-95060-5
Category:
Textbook
[Reviewed by
Steve Kennedy
, on
04/10/2001
]

This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. It might not be a good idea to create such expectations. You might not want to adopt this text unless you're comfortable teaching from a book in which the exposition will nearly always be clearer than your lectures. Understanding Analysis is perfectly titled; if your students read it, that's what's going to happen.

Teaching a one-semester introduction to real analysis can be a tricky business — on the one hand the entire point of the course is to construct, in excruciating detail, the theoretical underpinnings of calculus. On the other hand, going to such extraordinary lengths to prove theorems that the students already know to be true (and have never doubted) can seem to them a deliberate exercise in obfuscation. Let's face it, none of us has ever convinced a student that x2 is continuous with an epsilon-delta argument. Of course, we don't really make the argument for that purpose; our real purpose (usually unstated) is to convince them that our definition of continuity is reasonable — the average student usually misses this point entirely. The obvious solution to the dilemma is to present the problems and the counter-intuitive examples that informed and motivated the theory in the first place. One can be tempted, and I admit I was tempted, to attempt a historically accurate presentation of the subject. There exists a terrific book by David Bressoud, A Radical Approach to Real Analysis, published by the MAA, for those brave (or foolhardy) enough to take this radical approach. I think this is a mistake for a first pass through the theory and, if a student is only going to see this material once, she should get the cleaned-up, elegant, modern version. Future mathematicians, high-school teachers and the historically inclined can then be exposed to a historical treatment and be in a much better position to understand the evolution of the ideas if they are not also simultaneously trying to assimilate the technical details.

Thus a first course in analysis becomes a delicate balancing act between motivation and rigor. Steve Abbott's balance is nearly perfect. His text presents the standard topics of one-variable real analysis in the standard order. The distinguishing features are: the clear and easy prose style — this guy's writing is like a comfortable old shoe; the intuition-forward presentation — most concepts are tried out and walked around a little in the text before we get down to the nitty gritty; and the opening section of each chapter which presents a real problem that the rest of the chapter is designed to solve. Chapter two, infinite series, begins with a section on the pitfalls of rearrangement and asks what meaning can be attributed to a double summation. The chapter on continuity presents some (for us) old friends, Dirichlet's and Thomae's functions, and asks what sets can be sets of discontinuities of a function. Every chapter also concludes with i) a project section in which the chapter topic is explored more deeply but the proofs are only hinted at, filling in the missing details makes for a challenging exam or oral presentation project, and ii) a historical epilogue.

Of course no text is perfect and no mathematical book review is complete without quibbles: the standard proof of the chain rule is presented with no intuitive explanation, just a warning that it's a trick. The chapter on power series is motivated by a discussion of Francis Galton's use of branching processes. While interesting, this is somewhat ahistorical; I think Euler's computation of the sum of the reciprocals of the squares would be a much better choice (or, if that's too tired and overdone, maybe Newton's derivation of the binomial theorem and/or the arcsine series).

I should also mention that there are plenty of lovely exercises including many of a type I believe are particularly pedagogically potent, to wit: Construct an example of an object (function, sequence, series) that has property X (differentiability, convergence, absolute convergence), but does not have property Y (bounded derivative, reciprocals converge, absolute convergence when squared), or prove that no such object exists. (Oh, and a friendly tip — make sure you can do the exercise on the existence of uncountable anti-chains in the power set of N before you assign it, the author just might not respond to your panicked e-mail request for a hint before class meets!)

This terrific book will become the text of choice for the single-variable introductory analysis course; take a look at it next time you're preparing that class.

Steve Kennedy (skennedy@carleton.edu) is Associate Professor of Mathematics at Carleton College.

Preface v

1 The Real Numbers 1

1.1 Discussion: The Irrationality of 2 . . . . . . . . . . . . . . . . . 1

1.2 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The Axiom of Completeness . . . . . . . . . . . . . . . . . . . . . 13

1.4 Consequences of Completeness . . . . . . . . . . . . . . . . . . . 18

1.5 Cantor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Sequences and Series 35

2.1 Discussion: Rearrangements of Infinite Series . . . . . . . . . . . 35

2.2 The Limit of a Sequence . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 The Algebraic and Order Limit Theorems . . . . . . . . . . . . . 44

2.4 The Monotone Convergence Theorem and a First Look at Infinite Series . . 50

2.5 Subsequences and the Bolzano–Weierstrass Theorem . . . . . . . 55

2.6 The Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . . 58

2.7 Properties of Infinite Series . . . . . . . . . . . . . . . . . . . . . 62

2.8 Double Summations and Products of Infinite Series . . . . . . . . 69

2.9 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Basic Topology of R 75

3.1 Discussion: The Cantor Set . . . . . . . . . . . . . . . . . . . . . 75

3.2 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.4 Perfect Sets and Connected Sets . . . . . . . . . . . . . . . . . . 89

3.5 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.6 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4 Functional Limits and Continuity 99

4.1 Discussion: Examples of Dirichlet and Thomae . . . . . . . . . . 99

4.2 Functional Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 Combinations of Continuous Functions . . . . . . . . . . . . . . . 109

4.4 Continuous Functions on Compact Sets . . . . . . . . . . . . . . 114

4.5 The Intermediate Value Theorem . . . . . . . . . . . . . . . . . . 120

4.6 Sets of Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5 The Derivative 129

5.1 Discussion: Are Derivatives Continuous? . . . . . . . . . . . . . . 129

5.2 Derivatives and the Intermediate Value Property . . . . . . . . . 131

5.3 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . 137

5.4 A Continuous Nowhere-Differentiable Function . . . . . . . . . . 144

5.5 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6 Sequences and Series of Functions 151

6.1 Discussion: Branching Processes . . . . . . . . . . . . . . . . . . 151

6.2 Uniform Convergence of a Sequence of Functions . . . . . . . . . 154

6.3 Uniform Convergence and Differentiation . . . . . . . . . . . . . 164

6.4 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.5 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.6 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7 The Riemann Integral 183

7.1 Discussion: How Should Integration be Defined? . . . . . . . . . 183

7.2 The Definition of the Riemann Integral . . . . . . . . . . . . . . . 186

7.3 Integrating Functions with Discontinuities . . . . . . . . . . . . . 191

7.4 Properties of the Integral . . . . . . . . . . . . . . . . . . . . . . 195

7.5 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . 199

7.6 Lebesgue’s Criterion for Riemann Integrability . . . . . . . . . . 203

7.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.1 The Generalized Riemann Integral . . . . . . . . . . . . . . . . . 213

8.2 Metric Spaces and the Baire Category Theorem . . . . . . . . . . 222

8.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8.4 A Construction of R From Q . . . . . . . . . . . . . . . . . . . . 243

Bibliography 251

Index 253