I am a college sophomore in US with a major in mathematics and an aspiring mathematician in the computation theory and cryptography. I recently got an undergraduate research in the computation theory, which is heavily based on the algebra but also requires the extensive knowledge in the measure theory, approximation theory, and other bits of real analysis. On this upcoming academic year, I will be taking the year-long abstract algebra, linear algebra with proofs, applied linear algebra, combinatorics, and multivariable calculus (chug-and-plug level; taking on Spring 2016). I won't be taking any analysis course until next academic year. This means that I will be self-studying the real analysis by myself.
My research adviser recommended me to read the following books: Rudin (RCA), Folland (Real Analysis), Royden (Real Analysis), Stein-Shakarchi (Real Analysis), Cohn (Measure Theory), Halmos (Measure Theory), and Trefethen (Approximation Theory). If my memory serves me correct, those books are considered as advanced real analysis books that even touch a functional analysis. I told my adviser that I never took a mathematical analysis course, let alone any analysis book. He then advised me to self-study the real analysis, which I am motivated and excited to do so, and he devised a following self-studying plan for real analysis:
Learn the basics of analysis (real number system, basic topology, limit, continuity, and basics of series/sequence) from the elementary analysis books (such as Rudin's PMA, Apostol's Mathematical Analysis, and Pugh's RMA), and jump directly into the real analysis books mentioned on the last paragraph, and study them in a "non-linear, backward" style. That means that I will study those advanced real analysis books and learn the necessary topics from basic analysis books as needed. He said that method will allow me to learn the proper real analysis, gain deeper understanding, and prepare myself for the upcoming research and even Putnam competition. He even gave me an advise that the mathematics is usually learned in a nonlinear style, and it is always a good method to learn the materials from the advanced, comprehensive books.
I like his plan but I fear that perhaps it is too risky. Perhaps my adviser overestimated my ability. Do you like the plan? What is your suggestion? What other advanced real analysis books do you recommend to learn with what I mentioned? Will my lack of knowledge in the multivariable calculus be a significant problem to understand those texts?
MY BACKGROUND: I am currently studying and enjoying Artin's Algebra and Hoffman/Kunze's Linear Algebra.