Some hints and select solutions to the exercises can be found here. There are more to come.
I have received many requests from self-studiers for complete solutions to a handful of exercises from each chapter so that they can check their work; in response to this I do plan to provide some complete solutions. For many other problems I will provide hints.
I am working out a way to do this in such a way that the solutions do not come up in Google searches, as a number of exercises in my book also appear in others' books, and professors around the world have reasonable concerns about the proliferation of solutions online.
I am working on a timeline of the development of real analysis. You can see this timeline here.
To read historical papers and interesting modern expository articles, click here.
Many ideas in analysis are visual, and I've collect some awesome real analysis video from YouTube.
This book is the first of a (planned) series of textbooks which I am calling “long-form textbooks.” Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations.
The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics.
The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. In the first appendix we construct the reals, and the second is a collection of additional peculiar and pathological examples from analysis.
I also believe that learning mathematics has become too expensive and commercialized, and textbooks in particular are far overpriced. I am endeavoring to help change this. This 430 page book is currently available on Amazon for just $17, and in my own classes I offer a free alternative. The book can be purchased on Amazon here.
I am aware of thirteen universities which have used this textbook in their classes: Sacramento State University, Northeastern University, University of Nebraska--Lincoln, Portland State University, San Diego State University, CSU Long Beach, Humboldt State University, De La Salle University Manila, Roger Williams University, UC Santa Cruz, University of Houston, Grand Valley State University and State University of Ponta Grossa. If you know of another school which has used it, please let me know!
Traditionally, math classes were taught in the "sage on the stage" style, in which a professor lectures at students, keeping tightly to the material with little to no discussion. This is slowly changing, and research shows that it is changing for the better.
Proof-based textbooks are traditionally a dry list of definitions and results, with terse proofs of those results and not much else; "sage on the page," if you will. These textbooks are typically written by the foremost research experts in the field. I have great respect for these experts, and one of the most challenging tasks in writing a book is the determination of what to include and in what order to present it. These experts did a tremendous job on this, and without their work authors like myself would be unable to write the books we do.
That said, I believe a shift is beginning to occur in the textbook market. As prices continue to climb and mega-corporations are consolidating larger and larger slices of the industry, something has to give. Moreover, upper division math textbooks seem to be written for only the strongest and most self-motivated students. There are about 20,000 bachelor degrees given each year in mathematics and statistics at US institutions. And according to the AMS, there are only about 2,000 PhDs given in these fields per year. So even if one argues that reading terse math is proper because math research articles are (regrettably) written tersely, that argument ignores 90% of undergraduates for whom we want to understand and appreciate the material. Furthermore, it is always worthwhile to remind ourselves that only about half of the 10% that do go on to graduate school end up working in academia, so as pointed out by Stan Yoshinobu---professors are peculiar, and we shouldn't fall into the trap of only teaching to the students we were.
I have argued that there is a need for a new style of textbook, and fortunately I came to this realization just as the gatekeepers of publishing were critically weakened by the advent of self-publishing and online marketplaces. I am not an analyst. I really like analysis, have taken half a dozen semesters of analysis and have self-studied on top of that, but I have never published a single paper in the field nor do I claim to be an expert. A decade ago there is no way that Springer would approach someone like me to write an analysis textbook, however today Amazon's self-publishing arm couldn't be happier to print my book. A decade ago I would have had enormous difficulty bringing my book to the attention of others, but with nearly a hundred million Amazon users in the US and even more worldwide, the only gatekeepers are my readers (please rate and review!). I expect this reality to change the publishing market in the next couple decades, as more educators self-publish their own takes on the classic textbooks.
Another benefit of self-publishing through Amazon is that it better allows one to keep the costs down. Here's the publication timeline: