Companion Books for Rudin's PMA

Question

S.E friends,

I am a college sophomore with double majors in mathematics and microbiology. I wrote this email to seek your recommendation on selecting the introductory analysis textbook, particularly one that complement with Rudin's PMA well. Starting on this Fall, I will be taking Analysis I course, which uses Rudin's PMA. I will be finishing Calculus II (text: Calculus with Analytic Geometry, George Simmon; computational), and I also have been studying linear algebra (Serge Lang's Introduction to LA) and mathematical proof (Chartrand) book independently and will complete both subjects by mid-May. I was thinking studying Apostol/Spivak during Summer to prepare for Analysis I but I thought it would be best to just enter the analysis with Rudin's PMA and other analysis textbooks since it will be impossible to finish Apostol/Spivak during Summer and that time can be better spent on Rudin's PMA and others. I have enough money to purchase two other analysis textbooks that can complement Rudin's PMA well and help me to learn the analysis. My mind is on Apostol (Mathematical Analysis), Pugh, Ross, Strichartz, Lang, and Abbott, but I am not sure of their contents....unfortunately, those books have been either checked out or in hold at my university's math library. Please give me recommendation of two analysis textbooks that can supplement Rudin's PMA well!

Sincerely,

PK

Answer 1

I will strongly recommend a classic one : The Elements of Real Analysis, Second Edition: by Robert G. Bartle

Answer 2

Get a more intuitive book. Rudin is very overrated and provides little intuition. Plus, after chapter 8, the whole thing turns into one big dumpster fire. Forego hardcore delta-epsilon proofs and try to develop intuition. Apostol is better with that. Get used to being somewhat sloppy and get a handle on sending expressions to zero, making perturbation arguments, and building a solid toolkit for dealing with proofs.

After you've built up intuition, then try to tackle some of Rudin's exercises.