Analysis A good real analysis introductory book

You could try Understanding Analysis by Abbott. Another little known book is Analysis: With an Introduction to Proof by Steven Lay. You can order an older copy for relatively cheap from amazon.

 

You may also consider "Foundations of Mathematical Analysis" by Johnsonbaugh. It's a Dover book, so it's relatively cheap.

 

I didn't take analysis in college, but I did attempt to self-study a few years back. I initially tried to use Baby Rudin, which I wouldn't recommend (though it may be time for me to try it again since I'm in need of a refresher and the conciseness of this book may be what I need). You may want to check out samples for The Way of Analysis, which ended being the text I went through for my self-study. I enjoyed it. It is criticized for being the opposite of Baby Rudin, verbose, but it may be just what you need for a first pass on the subject.

 

I really like Rudin's PMA as it allows me to think in all directions to decipher the meaning of the exposition and figure out the connecting bridges between the concepts. It is definitely not an easy book to begin your first analysis, but PMA is a good book to improve your mathematical maturity and abstract thinking.

I also recommend Apostol's Mathematical Analysis and Buck's Advanced Calculus. If you would like to learn analysis with extensive use of the algebra, then I recommend the three volumes Analysis I-III by Herbert Amann.

 

I'm almost done with my first course in real analysis and we've been using Abbott's Understanding Analysis, and it's one of the best textbooks I've used (especially compared to my differential geometry course, where we're using Do Carmo... *shudder*). Abbott starts out with the axiom of completeness and builds off of that for every theorem that follows, which I really liked: it never feels like something is being pulled out of thin air.

If you aren't comfortable with proofs, check out Book of Proof:
http://www.people.vcu.edu/~rhammack/BookOfProof/
It's an introduction to the logic of proofs and how to use them, available for free online. I referred to it a bit when I was getting started with real analysis, and it's free, so it couldn't hurt to check it out. Also, the first chapter of Abbott is a very gentle introduction to set theory and mathematical logic, which could be very helpful. Good luck!

 

There is a book I really recommend for first-time and second-time studying of the real analysis. The book is called "Elementary Real and Complex Analysis? by G. Shilov. The book is at the level of Rudin's PMA, but it has clearer explanation and proofs (still terse but he provides important piece and logical progression so you can easily construct one by yourself). It is from Dover, so the price is very affordable too.