“PLEASE REMAIN CALM!”
So screams the introductory text to the structure of the real number system in this playful and delightful book on analysis.
Back in graduate school, the class to fear was real analysis. Fellow graduate students feared this course. They did remain calm. The class would weed-out (as we were told) the weaker students. If you survived it, you would learn how to do proofs. That was the key to real analysis — the construction of solid proofs built with logic in a step-by-step process from beginning to end. The class taught students to think and to think about all possibilities and not to just jump to a conclusion.
In this book, I saw the same goal. The material can be found in most textbooks on real analysis. What cannot be so easily found is the author’s presentation and guidance.
The book opens with a short but necessary discussion on logic, truth tables, and implications. It goes on to show the reader how to construct proofs by contradiction, a favorite for beginners, and delves gently into set theory. Once the reader is comfortable, Schramm discusses construction of a proof by the “forward-backward method.” This is a natural technique that most students come to on their own where the construction is done from both ends (beginning and end) and the result is to fill in the pieces of the proof from each end.
Some concepts of infinity are discussed such as cardinality of the natural numbers and rationals and, of course, the reals. Although a bit outside of the scope, I would have liked to see some discussion of transfinite numbers. Shramm takes the reader to the edge of infinity, but no further.
Having glimpsed infinity, we are treated to the arithmetic of reals, a study of fields, and the concept of orderings of sets. With ordering comes induction and this book presents this approach extremely well. Infinity comes again with the mention of transfinite induction but only to the edge. Schramm doesn’t explore this other than to show the reader the idea. Part one concludes with intervals and neighborhoods; continuity is not far behind.
Part two is about the structure of real numbers and we meet Theorem R that encompasses Nested Intervals, the Bolzano-Weirstrass theorem, Cauchy sequences, and the Heine-Borel theorem. The book does a remarkable job in tying these ideas together and each step is carefully presented with definitions, proofs, and examples.
We soon meet the topology of reals and see open sets and their structure which leads us to functions. Continuity here we come! The epsilon-delta proofs are next. These are given ample attention and left me with a deep sense of deja vu. Set theory ends with connected sets, and a bit more on continuous functions.
The last major part of the book is about calculus. The calculus discussion begins with series and trying to build integration from partial areas under a curve. The discussion here is excellent on Riemann integration and one can begin to see the types of functions for which Riemann integration fails. Like transfinite numbers, we only get a glimpse of what else is possible.
Uniform continuity and differentiation are next, followed by Taylor polynomials. Schramm presents Taylor series and does an admirable job explaining them and how an engineer, like me, could use them to approximate other functions. None of this is new but as above, the presentation is clear and well-done.
Calculus concludes, for the most part, with integration. The discussion is somewhat limited but the text gives the reader enough information to show where other work would lead. It is integration with details of what integration means, but without integral tables. Along similar lines, integration by parts is mentioned not to give one the formula, but to show just how deep is the connection of integrator and integrand. As I recall, freshman calculus courses only show the formula. We get a glimpse that there’s more to it, but we don’t get to see all that there is to it.
The examples are well done, with step by step guides to build proofs and introduce still more concepts. I liked the problems at the end of each chapter because they not only stressed what was already presented, but often introduced new concepts and guided the reader to develop them by himself.
This was a fun book, worth reading, and worth spending time exploring. But don’t remain calm, get excited. You’ll enjoy it.
David S. Mazel received his Ph. D. from Georgia Tech in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.