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An Invitation to Real Analysis

Luis F. Moreno
Publisher: 
Mathematical Association of America
Publication Date: 
2015
Number of Pages: 
661
Format: 
Hardcover
Series: 
MAA Textbooks
Price: 
75.00
ISBN: 
9781939512055
Category: 
Textbook
[Reviewed by
Mark Hunacek
, on
07/29/2015
]

This is an introduction to basic real analysis (single-variable theory only) intended as a text for a course that aspires, I think, to be at the easy end of the spectrum. The topics covered are (mostly) traditional, but the explanations given in the text are somewhat more detailed, and the pace of the development correspondingly slower, than in many other competing texts.

Instead of chapters that are then broken up into sections, this book is divided into 56 chapters and six appendices. The chapters vary in length; some are as short as four pages, others as long as 17 or 18. The text starts with a short introductory chapter 0 on paradoxes, giving some examples of counter-intuitive limit-based ideas that help motivate the need for a precise development of calculus. From here, there are a couple of chapters on logic and proof, followed by several on functions and basic set theory. Various number systems (natural numbers, rational numbers, and real numbers) are discussed, each by looking at axioms satisfied by that system. The axioms for the real numbers, not surprisingly, are those of a complete ordered field.

After this, there are several chapters devoted to sequences and convergence, including one on the lim sup and lim inf. These chapters lead naturally to several on limits and continuity of functions, which are then followed by several on infinite series. The numbers \(e\) and \(\gamma\) are defined (via limits), which leads to the definition of the exponential and natural log functions. Following this, there are chapters on differentiation, followed by several chapters on infinite sequences and series of functions, including uniform convergence and power series. The Riemann integral is then defined and studied for several chapters, including one on improper integrals. Following this, there is a chapter on the Cauchy-Schwarz and Minkowski inequalities (these are referred to later, but their appearance here seemed unmotivated and out of place). The remaining chapters deal with several topology-related issues (metric spaces, some real-number topology, and a chapter on the Cantor set).

There are then several appendices. One of these is just a list of symbols and Greek letters, but the others deal with technical issues that the author deemed too peripheral to the main text. An interesting feature here are the two chapters on Farey sequences and continued fractions, topics that rarely get discussed in basic real analysis courses.

As noted earlier, the explanations in this book tend to be somewhat more detailed and leisurely than in many other books on this subject. The sequential characterization of continuity, for example, which in most texts is done as a theorem, here merits an entire chapter. The definition of an infinite sequence, which usually takes just a few minutes of class time, is also the subject of an entire chapter here. Then, not only is there a chapter on the definition of the limit of a sequence, there is also one on what it means for a sequence to not be convergent.

Definitions are sometimes given in multiple, though of course equivalent, ways; for example, the statement that the limit of a sequence is the number \(L\) is defined first in terms of intervals (any interval centered at \(L\) contains all but a finite numbers of terms of the sequence) and then, immediately thereafter, in the usual “\(\varepsilon, N\)” way. This strikes me as a good idea; it helps the student understand what the definitions mean.

There are other features of the text that I think are valuable. There are lots of worked out problems in the text, for example, and also quite a few exercises. Some of these call for proofs, others are “advanced calculus”-type problems involving calculations with nontrivial examples. Solutions to the odd-numbered exercises appear in a lengthy (about 100 pages long) section of the text; a rather detailed solutions manual, containing solutions to problems of both parity, is available from the publisher to qualified instructors. The proof-based exercises struck me as being ones that most students should be able to do without too much difficulty.

Another nice feature is the greater-than-usual emphasis on history. The chapter on differentiation, for example, begins with a brief historical discussion, including an excerpt from Newton.

I also liked the fact that the author included an annotated bibliography (although I do think that any annotation to Bell’s Men of Mathematics should make clear that reliance on his factual assertions can be problematic), and also made use of articles appearing in journals like the College Mathematics Journal and Mathematics Magazine. One quibble: while these articles are cited in the text when used, they are not cited in the bibliography, which consists entirely of books. I also thought that there was an odd disconnect in listing journal articles for student perusal in a text where explanations are given in such detail; students who need to see multiple definitions of the limit of a sequence do not seem to be the optimal audience for reading articles in journals. However, I do like annotated bibliographies and I also like encouraging students to read journal articles, so I won’t cavil too much.

One other objection that I have, though, seems to me to be somewhat more substantial. Detailed explanations have obvious pedagogical value, but they do come at a price. This book is quite large (about 660 pages) and bulky, about twice the size of many competing texts. Appearance aside, it also takes a long time to get to the good stuff. For example, by the time the author has done all the standard material about the real numbers and is ready to define sequences, we are already 135 pages and 16 chapters into the text. Derivatives don’t appear until chapter 36 on page 319.

Of course, one might reasonably argue: “What’s the harm in having a book be too long or detailed? The more students that can understand it, the better.” One problem, I think, is that many students in a real analysis course are mathematics majors who are bound for graduate school, and these students need to get some experience reading mathematics at a fairly high level. A book that emphasizes hand-holding is very valuable in teaching the actual mathematics, but is less valuable when it comes to teaching the student how to read mathematics and cope with succinct explanations.

There is also, of course, the question of getting through the book. The author states in the preface that in a 15-week course, the entire text can be covered. If he says that he has done so, I of course take him at his word, but based on my own experience I doubt very much that I could pull that off. I taught real analysis in the fall 2014 semester, from a much more succinct text, and only managed to cover the real numbers, sequences and (very briefly) series, limits and continuity, differentiation and the Riemann integral. I never managed to discuss the concept of lim sup (I found that I never really needed it, and I suspected the students would find it very difficult), and also never got to sequences of functions and uniform convergence. I mentioned metric spaces only briefly, and didn’t spend as much time as I would have liked with the topology of the real numbers. I certainly never got to things like Euler’s constant \(\gamma\), improper integrals, or power series. In other words, I never got close to covering the entire contents of this text.

It seems to me that a professor using this book with any expectation at all of getting through all of it will have to make many decisions as to what to cover in class and what to assign as outside reading. Somebody who approached the book with the idea that the amount of time spent in class should be commensurate with the amount of pages spent in the text would probably never even reach differentiation as a topic.

To summarize and conclude: This is certainly a competent and well-organized presentation of the material; however, because I prefer something more succinct, I doubt that I would use this as a text for the introductory real analysis course here at ISU. However, it is a book that I would ask our library to put on reserve for possible optional perusal by students, particularly those who felt that they needed a slower and more detailed account of the material.


Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University. 

To the Student
To the Instructor
0. Paradoxes?
1. Logical Foundations
2. Proof, and the Natural Numbers
3. The Integers, and the Ordered Field of Rational Numbers
4. Induction and Well-Ordering
5. Sets
6. Functions
7. Inverse Functions
8. Some Subsets of the Real Numbers
9. The Rational Numbers are Denumerable
10. The Uncountability of the Real Numbers
11. The Infinite
12. The Complete, Ordered Field of Real Numbers
13. Further Properties of Real Numbers
14. Cluster Points and Related Concepts
15. The Triangle Inequality
16. Infinite Sequences
17. Limit of Sequences
18. Divergence: The Non-Existence of a Limit
19. Four Great Theorems in Real Analysis
20. Limit Theorems for Sequences
21. Cauchy Sequences and the Cauchy Convergence Criterion
22. The Limit Superior and Limit Inferior of a Sequence
23. Limits of Functions
24. Continuity and Discontinuity
25. The Sequential Criterion for Continuity
26. Theorems about Continuous Functions
27. Uniform Continuity
28. Infinite Series of Constants
29. Series with Positive Terms
30. Further Tests for Series with Positive Terms
31. Series with Negative Terms
32. Rearrangements of Series
33. Products of Series
34. The Numbers e and γ
35. The Functions exp x and ln x
36. The Derivative
37. Theorems for Derivatives
38. Other Derivatives
39. The Mean Value Theorem
40. Taylor’s Theorem
41. Infinite Sequences of Functions
42. Infinite Series of Functions
43. Power Series
44. Operations with Power Series
45. Taylor Series
46. Taylor Series, Part II
47. The Riemann Integral
48. The Riemann Integral, Part II
49. The Fundamental Theorem of Integral Calculus
50. Improper Integrals
51. The Cauchy-Schwarz and Minkowski Inequalities
52. Metric Spaces
53. Functions and Limits in Metric Spaces
54. Some Topology of the Real Number Line
55. The Cantor Ternary Set
Appendix A: Farey Sequences
Appendix B: Proving that nk=0<(1+1n)n+1
Appendix C: The Ruler Function is Riemann Integrable
Appendix D: Continued Fractions
Appendix E: L’Hospital’s Rule
Appendix F: Symbols, and the Greek Alphabet
Annotated Bibliography
Solutions to Odd-Numbered Exercises
Index