# Introduction to Real Analysis books

I am looking for a book that covers introduction to real analysis. Currently, I am reading The Elements of Real Analysis, by Robert Bartle. However, I quickly noticed that about half of the theorems and all of the sample questions don't have solutions to them so it's hard for me to know if my answers are correct so I looks around and was able to find the following book on the internet Principles of Mathematical Analysis which does provide a solution manual.

Comparing the two books, they do have some different topics so not sure what book what be best for me.

Are there any other highly recommend book which will be good for an introduction to analysis that provides a solution manual.

• Bartle's book has much better exercises than Rudin's, in my opinion. – Bungo Nov 28 '18 at 2:21
• If you want to learn the bare minimum analysis for higher mathematics. You have to learn from Foundations of Modern Analysis by Jean Diudonne. It, I am afraid, does not have solution manual; however, he does provide lengthy hints to the difficult questions. – Will M. Nov 29 '18 at 1:17

I would highly recommend the book by David Brennan called A first course in analysis (Cambridge University Press). What's good about it is it explains the definitions and walks you through all the "hard points" of the contents. The "other" books are also titled "intro" or "a first" course, but they are actually not so.

• I got the book and so far I really like it. It clearly illustrates how to work the examples and gives examples of the definitions instead of just giving the definition. Thanks! – Robben Dec 12 '18 at 23:24

Rudin's Principles of Mathematical Analysis is a hard book, but it's also a standard and it is extremely well-written (in my and many others' opinion) so you should read it early on. It may not be your first book in analysis, but if not I would make it your second.

When working through Rudin, even though you have a solutions manual, you should not give up on problems before you have solved them. There are problems in that book that take some of the best students hours over days to solve. The process of banging your head against the wall (or the book, or any other hard object) is part of the book and part of your preparation for mathematics. When you do get through Rudin, you will be in a very good place to step into the field of analysis $$-$$ possibly even the next Rudin book, Real and Complex Analysis.

As a soft introduction to analysis before Rudin, I would recommend my teacher's book: Mikusinski's An Introduction to Analysis: From Number to Integral. It is fairly short and easy to get through, and will prime your brain for the more intense fare of Rudin's book. It only covers single-variable analysis, however, which is 8 out of 11 chapters in Rudin; most courses in analysis only necessarily cover the first 7 chapters anyways.

Understanding Analysis by S. Abbott is a great introductory text. There is a lot of discussion, both informal to gain intuition and formal to rigorously pin down ideas. It felt like I was having a conversation the first time I read it, which was a very valuable experience as analysis was my first “hard” upper level class.

Edit: I do believe it has a solution manual as well.

See Introduction to Calculus and Classical Analysis by Omar Hijab. The synopsis of the book on the publisher's website is as follows:

"This text is intended for an honors calculus course or for an introduction to analysis. Involving rigorous analysis, computational dexterity, and a breadth of applications, it is ideal for undergraduate majors. This third edition includes corrections as well as some additional material."

Some features of the text: -The text is completely self-contained and starts with the real number axioms;

• The integral is defined as the area under the graph, while the area is defined for every subset of the plane;

• There is a heavy emphasis on computational problems, from the high-school quadratic formula to the formula for the derivative of the zeta function at zero;

• There are applications from many parts of analysis, e.g., convexity, the Cantor set, continued fractions, the AGM, the theta and zeta functions, transcendental numbers, the Bessel and gamma functions, and many more;

• Traditionally transcendentally presented material, such as infinite products, the Bernoulli series, and the zeta functional equation, is developed over the reals;

• There are 385 problems with all the solutions at the back of the text.

See too Elementary Classical Analysis by Jerrold E. Marsden and Michael J. Hoffman. I believe the 'worked examples' you'll find there can help you in the lack of answers to the exercises proposed in your self-study journey.

[Reviewed by Allen Stenger, on 11/25/2012 ]

"This is an introductory text in real analysis, aimed at upper-division undergraduates. The coverage is similar to that in Rudin’s Principles of Mathematical Analysis and Apostol’s Mathematical Analysis. This book differs from these earlier books primarily in being more talkative: explanations are written out at greater length, there are more worked examples, and there is a much larger number of exercises at all levels of difficulty."

Introduction to Real Analysis, by Bartle and Sherbert, appears to be the continuation to Elements. It also has an instructors manual, so you can check your solutions.

I used this book in college. It's a lot easier than Rudin's. If you are confident in your mathematical abilities, I would recommend Rudin's Principles of Mathematics. Don't beat yourself up if you can't get through it, though. It is terse.