Okay, here's the deal: I am going to run out of math classes at my high school at the end of this [junior] year. I'll have finished AP Calc BC (with Larson/Edwards as the text), but I already studied almost all of Gilbert Strang's free online text. So I'm really bored redoing material I already know. I want to do a rigorous self-study next year (maybe starting this year if I stay this bored). I want to work with a book that emphasizes theory and proof. I love the theoretical aspect and it would be great to get a taste of "real" math before heading off to college as a potential applied math major. My background is fairly standard USA K-12 math, with more rigorous intro calculus thanks to the Strang self-study. My requirements for a book: 1) It must emphasize proof and have challenging material. 2) I would prefer if explanations are somewhat readable - über-formal prose can be frustrating. 3) Problem sets are a must, since I need to work through exercises to understand material properly. The big three calculus/intro analysis texts (Spivak, Apostol, and Courant) are all obvious possibilities. Unfortunately, they are all really pricey. I could probably swing a copy of Spivak, but all those cheap Dover books have spoiled me. What would you suggest?