Basic Analysis: Introduction to Real Analysis

By: Jiří Lebl (website #1 (personal), website #2 (work: OSU), email:

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This free online textbook (e-book in webspeak) is a one semester course in basic analysis. This book started its life as my lecture notes for Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in the fall semester of 2009, and was later enhanced Sample Darboux sums to teach Math 521 at University of Wisconsin-Madison (UW-Madison). A prerequisite for the course is a basic proof course. It should be possible to use the book for both a basic course for students who do not necessarily wish to go to graduate school, but also as a first semester of a more advanced course that also covers topics such as metric spaces.

The standard book used for the class at UIUC is Bartle and Sherbert, Introduction to Real Analysis third edition (BS from now on). The structure of the notes up to chapter 6 mostly follows the syllabus of UIUC Math 444. Some topics covered in BS are covered in slightly different order, some topics differ substantially from BS and some topics are not covered at all. For example, we will define the Riemann integral using Darboux sums and not tagged partitions. The Darboux approach is far more appropriate for a course of this level. Chapter 7 (Metric Spaces) was added to teach the more advanced course at UW-Madison. The philosophy is that metric spaces are absorbed much better by the students after they have gotten comfortable with basic analysis techniques in the very concrete setting of the real line.

The aim is to provide a low cost, redistributable, not overly long, high quality textbook that students will actually keep rather than selling back after the semester is over. Even if the students throw it out, they can always look it up on the net again. You are free to have a local bookstore or copy store make and sell copies for your students. See below about the license.

One reason for making the book freely available is to allow modification and customization for a specific purpose if necessary (as the University of Pittsburgh has done for example). If you do modify this book, make sure to mark them prominently as such to avoid confusion. This aspect is also important for longevity of the book. The book can be updated and modified even if I happen to drop off the face of the earth. You do not have to depend on any publisher being interested as with traditional textbooks. Furthermore, errata are fixed promptly, meaning simply that if you teach the same class next term, all errata that are spotted are already fixed. No need to wait several years for a new edition.

MAA recently published a review of the book (they looked at the December 2012 edition). Plot of sin(1/x)

Table of contents:

1. Real Numbers (1.5 new 5/29/13)
2. Sequences and Series (2.6 new 5/29/13)
3. Continuous Functions (3.5, 3.6 new 5/29/13)
4. The Derivative (4.4 new 5/29/13)
5. The Riemann Integral (5.4, 5.5 new 5/29/13)
6. Sequences of Functions
7. Metric Spaces

There are 383 exercises (December 16th 2014 edition).

Please let me know at if you find any typos or have corrections, extra exercises or material, or any other comments. I will always keep all older versions available for download, at least when there are nontrivial updates. When the updates are reasonably minor, I will try to preserve pagination and numbering of sections/examples/theorems/equations/exercises as much as possible.

There is no solutions manual for the exercises. This situation is intentional. There is an unfortunately large amount of problems with solutions out there already. Part of learning how to do proofs is to learn how to recognize your proof is correct. Looking at someone else's proof is a far less effective way of checking your proof than actually checking your proof. It is like going the gym and watching other people work out. The exercises in the book are meant to be a gym for the mind. If you are unsure about the correctness of a solution, then you do not yet have a solution. Furthrermore, the best solution for the student is the one that the student comes up with him(or her)self, not necessarily the one which the professor or the book author comes up with.


Do let me know ( if you use the book for teaching a course! The book was used, or is being used, as the primary textbook at (other than my courses at UIUC, UCSD, and UW-Madison) University of California at Berkeley, University of Pittsburgh, Vancouver Island University, Western Illinois University, Medgar Evers College, San Diego State University, University of Toledo, Oregon Institute of Technology, Iowa State University, California State University Dominguez Hills, St. John's University of Tanzania, Mary Baldwin College, Ateneo De Manila University, University of New Brunswick Saint John, and surely others I do not know about.

The book has been selected as an Approved Textbook in the American Institute of Mathematics Open Textbook Initiative.

See a list of classroom adoptions for more details.


Download the book as PDF
(December 16th, 2014, 243 pages, 1.2 MB download)

Several new sections were added in the May 29th 2013 version. See the change log page to see what is new.

Check for any errata in the current version.

Look at the change log to see what changed in the newest version (there you can also download old versions if you wish).

Buy paperback:

I get a bit of money when you buy these (depending on where exactly they are bought, most if it is bought at the createspace store). Probably enough to buy me a coffee, so by buying a copy you will support this project. You will also save your toner cartridge. The difference between these two versions is essentially just the cover art. I have seen printed versions from both and they are both good quality.

Basic Analysis Cover
Buy a paperback copy at for $11.90. You can also buy a larger (letter size) coil bound copy at for $14.41. This copy is the December 16th, 2014 revision. No ISBN for the lulu version.

Basic Analysis Cover
Buy a copy at for $11.90, or buy a copy on Amazon. This is a paperback version on Crown Quatro size (7.44x9.68 inch). This should be a little easier to buy and quicker/cheaper to ship. This copy is the December 16th, 2014 revision. This one has an ISBN:
ISBN-13: 978-1505695915
ISBN-10: 1505695910


LaTeX source as a tarball. The main file is realanal.tex. I compile the pdf with pdflatex. You also want to run makeindex to generate the index (I generally run pdflatex realanal three times, then makeindex realanal, and then finally pdflatex realanal again).

During the writing of these notes, the author was in part supported by NSF grant DMS-0900885 and DMS-1362337.


Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. You can use, print, copy, and share these notes as much as you want. You can base your own notes on these and reuse parts if you keep the license the same. If you plan to use these notes commercially (sell them for more than just production cost), then you need to contact me and we will work something out. If you are printing a course pack for your students, then it is fine if the copy service or bookstore is charging a fee for printing and selling the printed copy. I consider that production cost.

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