Want advice for independent study books on introductory analysis? I'm taking an intro to analysis course in the fall, and due to COVID am quite bored...looking to learn and practice the material ahead of time.
Does anyone know any good (and preferably free) Intro to Real Analysis books, that have good practice problems with solutions?
Thanks
 A: I would suggest Analysis With An Introduction to Proof by Steven R. Lay.
It's the book I used to prep for my intro to real analysis class and it was better than the book my professor actually used for that class. It has good exercises and great descriptions. I will include a link. You should be able to find a used copy for a reasonable price.
https://www.amazon.com/Analysis-Introduction-Proof-5th-Steven/dp/032174747X/ref=sr_1_4?dchild=1&keywords=lay+intro+real+analysis&qid=1589156491&sr=8-4
A: I would like to suggest two books over here. 


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*Modern Calculus & Analytic Geometry by Richard Silverman


This is a fantastic text that emphasizes rigor over and over again. It contains proofs to almost all theorems and you'll get quite a lot of practice since each section has an entire problem set dedicated to it. There are 115 problem sets that contain many interesting and hard problems.
I should say that this particular author has also been responsible for the translation of many Soviet mathematics textbooks. I've used a couple of them and I can only say good things about them. He has used a couple of those sources in creating this amazing textbook. 
This book approaches Calculus very rigorously, much like Calculus by Michael Spivak does, so you can expect to learn quite a lot from it. You can expect yourself to become extremely familiar with many of the ideas from Analysis if you work through it patiently. 


*Mathematical Analysis by Bernd Schroder


When I was covering the single-variable part of the above-mentioned textbook, I often referred to this particular textbook for additional insights. More specifically, I would refer to this textbook if there was something discussed in Silverman's textbook that I wanted to know more about. For example, the limits of sequences is definitely covered in more depth over here than in Silverman's textbook. 
This text is very short compared to the one above. It covers a large, large amount of Analysis in slightly more than 500 pages. It covers just about everything from very basic notions of convergence of sequences to basic measure theory. 
So, it covers way more than you actually need but I think that you can definitely benefit from it. It has almost no prerequisites other than just a slight bit of mathematical maturity. If you're not comfortable with proofs, you can certainly get comfortable with them using Silverman's book before moving on to this one.
Because I used this as a reference quite a bit, I've read a sizable chunk of the first 13 chapters and I certainly enjoyed reading through them. Keep in mind that it is difficult and he does introduce the Lebesgue Integral in there, right after the Riemann Integral. So, you'll have to put in some effort to really understand everything but I think that it will be extremely helpful to you in preparing for the coming course and for future courses. 
If you have any questions about both books, please let me know and I will try my best to answer those questions. 
