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I am a sophomore in US with double majors in mathematics and microbiology. I am interested in self-studying the real analysis starting now since it will help me on my current research on computational microbiology, prepare for upcoming math research (starting on this Fall) on the analytic number theory, and prepare for the real analysis course I will take on 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 Calculu Vol.I to get a deeper view on the calculus theories. I was originally planned 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 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 PMA (required textbook for my real analysis course) starting on 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)

Thank you very much for your time, and I look forward to your advice!

Sincerely, PK

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    $\begingroup$ This ain't the physics forum, just so you know. $\endgroup$ – Mnifldz Mar 10 '15 at 20:22
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    $\begingroup$ Pugh is "detailed" and not at all "dumbed down", and on the same level as Rudin - as indeed is Apostol (which is great, bordering on essential). $\endgroup$ – Calum Gilhooley Mar 10 '15 at 20:33
  • $\begingroup$ I wouldn't say they're on the same level as Rudin...perhaps $\epsilon$ easier. ;) $\endgroup$ – Daniel W. Farlow Mar 10 '15 at 21:00
  • $\begingroup$ I am thinking either Lang or Apostol since I understand them...are they on the same level as Ross, Strichartz, Pugh, and Abbott? $\endgroup$ – MathWanderer Mar 10 '15 at 21:24
  • $\begingroup$ abbott and zorich... both are wonderful $\endgroup$ – Bhaskar Vashishth Feb 9 '16 at 20:26
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Other than Rudin's analysis text (the first one), I've read Robert Strichartz's "Way of Analysis". Strichartz gives you a lot of motivation and information for most of the concepts, while Rudin just gives you enough for you to do on your own. I favor Rudin's text much more, since I enjoyed the effort required to fill in the gaps and thereby making you "do" a lot during the reading. Additionally, Rudin has a style that I believe many enjoys, for most of his proofs are elegant and stylish. Strichartz will explain much more during, before, and after a proof, but sometimes I feel that his explanations becomes a bit too wordy and too cloudy over the main point. Take it this way: Would you rather try to find a dull sapphire in a messy hay sac, or find a beautiful diamond hiding in a small pile of needles? (one requires sheer effort for something "okay", while the other requires much more effort and care, but to obtain something rather nice.)

I realize that you want to read a more friendly analysis text before Rudin's, but have you considered reading Rudin's and supplementing it with one of the texts you've mentioned simultaneously? If you read Rudin and find that you have a lot of questions, then use the supplements and this site; questions will only help you.

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    $\begingroup$ Thank you very much for your advice! I have Rudin's PMA but I honestly cannot understand his book for now due to his terse writing style. Right now, I have been reading both Lang's Undergraduate Analysis and Apostol's Mathematical Analysis, and I actually understand what they say. My plan is to start with either of them and later (after one or two months) use Rudin with either of it (after I learn some basics). $\endgroup$ – MathWanderer Mar 10 '15 at 21:22
  • $\begingroup$ If I understand Lang and Apostol, do I need to look through Ross, Pugh, Strichartz, or Abbott? Also which one does cover more topics, Lang or Apostol? It seems that Lang does not cover the topology and I am worried it might affect the understanding of real analysis, but I am not sure. $\endgroup$ – MathWanderer Mar 10 '15 at 21:22
  • $\begingroup$ Also is it recommended to keep continuing with the proof methodology books? I heard that the best way to sharpen the proof skills is to actually solve problems rather than reading the proof books. $\endgroup$ – MathWanderer Mar 10 '15 at 21:25
  • $\begingroup$ @MathWanderer I haven't read those texts, so I can't answer that question. All I can say is that reading Strichartz's book with a basic understanding of proofs sufficed for me to read Rudin's text. However, based on what I've seen around the web, it appears that Apostol does have a reputation of intermediate difficulty, so I'm somewhat sure Strichartz shouldn't be a problem for you if you understand Apostol. If you're in a rush to reach Rudin, then I think it's sufficient to simply have Apostol's text under your belt. $\endgroup$ – Metric Mar 11 '15 at 2:46
  • $\begingroup$ @MathWanderer You should remember though that it's not wise to rush mathematics, but since you're about to have a course using Rudin, then I suppose this is a good option. And yes, doing proofs after having a sufficient understanding of them should definitely help, for doing any type of Mathematics greatly increases your understanding, much more than simply gazing through theorems and definitions all day. $\endgroup$ – Metric Mar 11 '15 at 2:48

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