Any suggestions for a textbook to self-study Real Analysis

Question

I'm currently in secondary school and I am looking to self-study some introductory real analysis, I have studied the prerequisites on MIT OCW but they have no real analysis stuff so I was hoping someone could suggest a textbook for me. Thanks in advance.

Answer 1

Personally I prefer Robert G.Bartle. You can go through the book in the form of PDF which is available on the Internet. Its quite comprehensive and concise. Just give it a go.

Answer 2

Disclaimer: I think this is a beautiful thing to do, especially if you aren't yet at the university.

First of all I think that you should revise what is usually meant by set theory (and "mathematical logic"); no axiomatic for the moment, but enough stuff to start having a feel for what does it mean to do proofs (a working knowledge of sets is a must for whatever mathematics you will meet from now on): this will help when you'll get stuck facing proofs that use such tricks, De Morgan's laws, ecc...

Even if it is not a Real Analysis textbook, I like Munkres, Topology: I think the first chapter will do its job, with respect to sets and logic. Note that usually - at least here in Italy - Real Analysis textbooks have, in the first chapters, an introduction to all this stuff (Baby Rudin excluded).

That being said, I think Baby Rudin is the best place to start if you want to explore the world of (elementary) analysis (but please also look at other people's question/answers: MSE is full of resource recommendation questions): its treatment of the subject is "topological", in a sense which will be more clear to you later, starting from the definition of metric space and developing the theory of continuous functions inside metric spaces (which, though being more abstract, covers several special cases at once, preventing you from understanding them as separate things); and learning a bit of topology simultaneously with analysis puts in your hands a number of useful tools also for other branches of mathematics. (Be anyway prepared to be discouraged by very terse proofs, which going through (by yourself as much as you can, and asking for help only when you have a solution to check) is essential to learn learning the subject.)

Consider also J. Dieudonneé, Foundations of Modern Analysis: in the first chapters there is more or less what you need to start doing analysis, and, even being an extremely terse book sometimes, it's a good read to do in parallel with Rudin.

Last, but not less important: do as much exercises as you can, in particular proofs over mechanical busywork.