[synacksynack]

Michaek Spivak's "Calculus" is pretty great for learning real analysis. The other canonical text is "Principles of Mathematical Analysis" by Walter Rudin, which I've never used, but is purported to be of the same quality.

[joe_the_user]

Actually,

I wouldn't recommend either of those two book for self-teaching.

For whatever reason, I taught myself advance mathematics in High Schools. Spivak and Rubin were pretty inaccessible. Sure, they are rigorous and high level but that meant they didn't lend themselves to an easy read. "Real Analysis" by Royden was relatively quick to go through - and I had only the start of calculus. Royden gives a fairly simplistic and accessible development of higher mathematics starting with set theory.

I'd imagine that would be the most helpful.

(In my High School years, I went from algebra sophomore year to reading Royden and doing a community college calculus class junior year to passing a differential geometry course and the undergraduate honors seminar at UCLA).

[zackattack]

op, i don't know your mind but i agree, Rudin/Spivak may prove challenging, especially without the help of a professor/TA to answer your questions in real life... i would not go and buy them blindly. the best course of action you could take would be to go visit a college bookstore and sit down with one of the analysis texts, and then read through it for a few hours. if you feel comfortable with it and don't experience any contempt for the author for introducing terms without explaining them, etc., then go ahead and get it. i would recommend finding something that holds your hand through the new fancy notation of proofs (for all, exists, epsilon, blah blah blah)...something that combines a basic intro to topology/set theory in its introduction

[henning]

Rudin is deceptive. At UCSD it takes a full academic year to get through that book. It is damn hard. UCSD was recently ranked in the top 50 universities in the world so it is not some lousy lazy place.

If you're going to try to learn that stuff on your own you'll need other books to supplement it and provide examples, counter-examples, intuition, and alternative explanations.

[jedbrown]

It's an undergrad text so it can't assume a great deal of mathematical maturity, hence everything takes longer.