## Arthur Mattuck: Introduction to Analysis

Massachusetts Institute of Technology

The book was developed at MIT, mostly for students not in mathematics having trouble with the usual real-analysis course. It has been used at large state universities and small colleges, as well as for independent study. Students evaluate it as readable and helpful. The current printing, by CreateSpace and at a reduced price, is the eighth, incorporating all known significant corrections and a new Appendix F.

### General description

This book is meant for those who have studied one-variable calculus (and maybe higher-level courses as well), generally skipping the proofs in favor of learning the techniques and solving problems. Now they are interested in learning to read proofs, and to find and write up ther own: perhaps because they will need this for the next steps in their chosen field, or for intellectual satisfaction, or just out of curiosity.

There are two paths to this. Some books start with a great leap forward, giving the definitions in n-space. This requires first an excursion into point-set topology, whose proofs are unlike those of the usual calculus courses and are a roadblock to many.

The path chosen by this book is to start like calculus does, in 1-space (i.e., on the line) and focus on the basic definitions and ideas of one-variable calculus: limits, continuity, derivatives, Riemann integrals, and a few more advanced topics. It's done rigorously, but also in as familiar a way as possible. So from the start it will use as a source of examples what you know (with occasional reminders): K-12 mathematics and basic one-variable calculus, including the log, exp, and trig functions. This takes up about two-thirds of the book, and might be as far as you wish to go. It sounds like this is just repeating calculus, but students say that it feels very different and is not all that easy.

The rest of the book gets into techniques from advanced calculus based on the notion of uniform convergence, and usually used in lower-level courses without proof: differentiating infinite series term-by-term, and differentiating integrals containing a parameter (the Laplace transform, for instance). For the latter, it's finally time to learn about point-set topology in the plane (i.e., 2-space, but n-space is no harder). There's also for the curious or needy an optional chapter with the most important facts about point-sets of measure zero on the line, and a more powerful integral: the Lebesgue integral.

Two appendices respectively provide needed and optional background in elementary logic, and four more give interesting applications and extensions of the book's theory.

--- Attention paid to layout and typography, both for greater readability, and to give readers models they can imitate;
--- Questions after most sections of a chapter to firm up what you just read, with Answers of various sorts at the end of the chapter: single words, hints, complete statements, formal proofs.
(See below for the link to Sample Text Pages to see examples.)

--- The language of limits is simplified by suppressing the N and the delta when their explicit value is not needed in the argument, replacing them with standard applied math symbols meaning "for n large" and "for x sufficiently close to a". These are introduced carefully and rigorously; some caution is needed, which is described at the end of the Preface (see the link below to it).
--- The book tries to go back to the roots of real analysis by emphasizing estimation and approcimation, which use inequalities rather than the equalities of calculus, but have a similar look, so that many proofs are calculation-like "derivations" that seem familiar. But inequalities require more thought than equalities; they are often mishandled and warnings have to be given and repeated.

### Looking at the book

For more details about what's written above, you can use these links in order to get an idea of how it's written, and what studying analysis from it will be like.

#### PREFACE: pdf file

Its five-page Preface gives some idea of the book's origins and style, and toward the end more details about the mathematically helpful features..

What's in the book (and by omission, what's not), and the order of topics.

#### SAMPLE SECTIONS: pdf file

These are a few sections, totalling about 15 pages in all, showing text material, Questions, Exercises, and Problems, to give you a sample of the mathematical writing style and level. They are selected from the first three chapters:
--- Chapter 1: Real Numbers and Monotone Sequences
--- Chapter 2: Estimations and Approximations
--- Chapter 3: The Limit of a Sequence

#### THE FIRST THREE CHAPTERS AND APPENDIX A:

These are posted here complete to accommodate students who will not have the book by the first class meeting.
Chapter 1: pdf file
Chapter 2: pdf file
Chapter 3: pdf file
Appendix A: pdf file

PRINTING: There is only one edition so far, but several printings. The printing is identified by a number sequence like 10 9 8 7 6 5 4 on the left-hand page facing the dedication page; the sequence shown identifies the fourth printing, for example.

CORRECTIONS: The current inexpensive printing is the eighth; it incorporates all the significant mathematical corrections needed from earlier printings.
I would be grateful to hear about any further corrections needed. as teacher or student; write to: mattuck@mit.edu

For those using printings earlier than the eighth, here are two lists of corrections. The first is exclusively to the problems: this includes the Exercises and Problems at the end of each chapter, as well as the Questions at the end of each section, and the Answers to those Questions at the end of each chapter.

#### Corrections to the problems in Printings 1-7 ( pdf file )

This second list of corrections is for the text material, i.e., everything that's not an Exercise, Problem, or Question and Answer). Only significant mathematical material is on the list. (The current eighth printing also tries to correct typos, misspellings, poor spacing, etc.)