I am a sophomore in the US with double majors in mathematics and microbiology. I am interested in self-studying real analysis since it will help me with my current research in computational microbiology, prepare for upcoming math research (starting this Fall) on analytic number theory, and prepare for the real analysis course I will take this Fall and Putnam competition.
I just finished Calculus with Analytic Geometry by G. Simmons, How to Prove It by Daniel Velleman, and How to Solve It by G. Polya. I also read some portions of Apostol's Calculus Vol. I to get a deeper view on calculus theories. (I was originally planning to read Apostol's Calculus Vol. I and Spivak's Calculus first, but I think it would be a better idea to start with real analysis since it covers all the ideas in those "advanced calculus" textbooks and much more.)
My current plan is to start with one "dumbed-down" real analysis textbook and one "comprehensive, detailed, and intermediate" textbook, and advance into Rudin's Principles of Mathematical Analysis (required textbook for my real analysis course) starting this Summer, and use it in accordance with other real analysis textbooks. Could you help me on selecting one book from each category?
Elementary Real Analysis textbooks:
- Elementary Analysis: The Theory of Calculus (Kenneth Ross)
- Understanding Analysis (Steven Abbott)
- The Way of Analysis (Robert Strichartz)
- Real Mathematical Analysis (Charles Pugh)
Intermediate, detailed Real Analysis textbooks:
- Mathematical Analysis (Tom Apostol)
- Undergraduate Analysis (Serge Lang)
- Introduction to Real Analysis (Bartle, Sherbert)
- Elements of Real Analysis (Bartle, Sherbert)
- Mathematical Analysis I (Vladimir Zorich)