An Invitation to Real Analysis exists to serve a variety of introductory courses in real analysis in a pedagogical interval, from a one-semester analysis-based introduction to proofs course near the left endpoint, to a one- or two-term course for more advanced students in mathematics and other quantitative fields at the right endpoint. The book is far from alone among analysis texts, so a few words are in order about the book’s features.
In my experience, real analysis tends to be the course of “maximum dynamic pressure,” a blizzard of complicated and unfamiliar definitions, unstated idioms involving the use of arbitrary inequalities to establish equalities, and bewildering technicalities stated as tone poems written from Greek alphabet soup. In effect, students are asked to read and write literature while learning a foreign language. Neglecting to mention this often creates a mysterious layer of frustration: Students who enjoyed calculus suddenly find their day-to-day understanding of mathematics has sharply decreased for no apparent reason.
Two pedagogical principles stand out for shaping the book. First, presenting definitions of analysis as adversarial games, not as a passing remark but as a uniform stance, powerfully engages students’ intuition. Second, real analysis is a language in addition to its substantial mathematical content. In particular, consistency of definitions is vital given the founding role analysis plays in so much of school mathematics, not to mention calculus and beyond.
Atop these anchoring principles, the book takes special care:
- To be self-contained (except for the integer division algorithm);
- To identify Greek letters on first usage and explain how to read (enunciate) expressions;
- To use consistent and evocative terminology, notation, and proof idioms, stringently avoiding unstated edge cases;
- To phrase definitions and theorem statements for ease of comprehension and use;
- To structure arguments as simply and concretely as possible, and only directly or contrapositively, not by contradiction;
- Starting with axioms for the real numbers in Chapter 3, and earlier where feasible, to develop material in strict logical order, with proofs resting solidly on definitions and specific, established theorems;
- To present interesting examples as soon as possible;
- To provide students with comparable proofs before posing exercises, both as illustrations of proof-writing and as incentives to study and absorb existing proofs;
- To encapsulate technical points to avoid repetition;
- To cultivate conceptual revisiting, foreshadowing coming ideas and connecting material with what has already been developed.
Terminology, Notation, and Conventions
We mathematicians tend toward eponyms for concepts and theorems. These are opaque to students, and rarely withstand historical inspection. With two peripheral exceptions that are well-known in popular culture—Venn diagrams and Fibonacci numbers—this book instead uses descriptive names: the ordered product of sets, condensing sequences whose terms can be made as close to each other as we like, the cross-term bound for inner products, the polar formula \(e^{i\theta} = \cos\theta + i\sin\theta\), spectral decomposition for periodic functions, and many others. Eponyms are, however, indexed for easy reference.
Students are not unlikely to have programming experience. A couple of expository choices reflect this reality. Definitions, theorems, and exercises are formulated with an eye toward algorithmic implementation where appropriate. Sums and factorials, for example, are defined recursively. Generally, the book avoids using ellipses to connote “continuing patterns” except for illustration. Consequently, mathematical induction underpins the entire book. More than once, these recursive foundations simplified the “traditional” presentations and proofs I had used for many years in the classroom.
In this book, curly braces signify unordered sets while round parentheses connote ordered lists. Counting starts at \(0\) (the cardinal of the empty set) and, for a list of length \(n\), ends at \((n – 1)\), as in programming languages. In this book, \(0^{0} = 1\), the number of mappings from the empty set to itself. I did not set out to make iconoclastic choices, but found while writing that being systematic with these conventions pleasantly clarified edge cases and simplified at least a few calculations.
Aspirations and Easter Eggs
An Invitation to Real Analysis is written for the 21st century. To the best of my ability, the book is a gradual, paved, well-marked path with multiple trailheads into the language and landscape of real analysis. A few fun ponderings, cultural references, and wry jokes are sprinkled throughout. I hope and expect both students and you will find the book especially friendly, supportive for your respective needs, internally consistent, mathematically substantive, and thought-provoking in positive ways. It is written to be read, studied, taught from, referenced for many years, and enjoyed.