A kind introduction to mathematical analysis by Daniel C. Bastos
Content: better than 99%
Writing: better than 99%
This is truly a beautiful book. I'm sure there are good books on Analysis out there, but take for instance, Principles of Mathematical Analysis, by W. Rudin; it is indeed a wonderful book, but you must have a real solid background to be able to enjoy it. Zakon's ``Mathematical Analysis I'' will show you how easy somethings can be by presenting the material in a nice, kind and very clear way with examples and everything you could expect to get a solid background on the subject. Of course, this isn't the only book one should study, but if most of what read on Analysis' books seem blurry, try this one out. I strongly recommend.
Zakon's Mathematical Analysis is divided in two volumes; I have not yet seen volume II (it is still in preparation), but by reading volume I you can obviously expect the same degree of didacticism on volume II. The chapters are laid out to benefit the reader's learning process and this is an important matter. I highlight the simplicity in which the authors treats every piece of the book. Zakon's Mathematical Analysis achieves a good degree of formality but still gives you the chance to learn what is being said. If you study this book, you will get the vocabulary you may lack on formal mathematics, you will get the background you need to read almost every other book on Real Analysis.
Another important thing that should be noticed is the license of this book. While institutions tell each student of Analysis to buy a fairly expensive book out there, they could be using one of great value for much less. As the license states, $300 would pay for the book for the entire institution. If you like this, drop a letter to your department and let them know about it.
The above review is copyrighted by its author, and is copylefted under the following license: GFDL 1.1
Rethinking the Elementary Real Analysis Course
By Brian S. Thomson, in the American Mathematical Monthly, Jun/Jul 2007, pp. 469-490.
We are now more than a century past the appearance of Lebesgue's paper announcing his integral. Yet our calculus and introductory real analysis courses continue to present a full treatment of Riemann's integral with no better justification than that the alternative might be too difficult for the students. This is not so. The correct integral on the real line can be introduced at an early stage, requiring no more preparation than a thorough treatment of compactness arguments. This article outlines some of the concerns that a designer of an elementary real analysis course should confront in developing an adequate theory of integration.