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When I first decided to begin real analysis I had a look at the much recommended book 'Principles Of Mathematical Analysis' by Rudin, however I found it very complicated and understood very little of it. I instead chose to use a combination of Ross and Bartle to learn from and it has been much clearer. I decided yesterday to have another look at Rudin's book and I seem to understand it a lot better and can follow it. The only difference is that the books I'm currently using don't use metric spaces. Should I stop using the ones that I am and just use Rudin, or should I finish Ross and then move on to Rudin afterwards for a more advanced treatment.

Should be noted that I am self teaching this subject.

J. W. Tanner
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Maximus
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    I’d say never limit yourself to just one book. Look at a bunch of books, and focus on whichever one connects with you. You should keep looking at Ross and throw some other books in the mix too. – littleO Jul 31 '23 at 12:39
  • @littleO Ok thanks. I was just wondering because Rudin covers everything in terms of metric spaces, which is slightly different to what I have been doing (currently up to BW theorem in sequences). My original approach was to cover Ross, and then to reach a more advanced level use the book 'Introduction to metric and topological spaces' by Sutherland. Im just not sure if I would be missing something by not using Rudin. – Maximus Jul 31 '23 at 12:42
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    I think you should look at Rudin also, and eventually you should probably study the relevant chapters of Rudin carefully, but for the time being you might find that you get more out of other books. Personally, I find baby Rudin to be tough to learn from. It is like an elegant repository of perfect, concisely written definitions and theorems and proofs; a work of art perhaps but it doesn’t waste many words on conveying intuitive understanding. – littleO Jul 31 '23 at 12:55
  • @littleO I appreciate you taking the time to respond given that this is a very vague question. – Maximus Jul 31 '23 at 12:57
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    Regarding @littleO's comment "Look at a bunch of books", see the first paragraph of this answer. There I discuss topology books, but the first paragraph pretty much applies to any academic topic (not necessarily math). Regarding the issue with metric spaces, I wouldn't worry about that now. At this point you're mostly learning rigorous definitions (e.g. open set, various formulations of continuity, etc.) and precisely stated theorems (mean value theorems, interchange of limit operations, etc.) (continued) – Dave L. Renfro Jul 31 '23 at 13:11
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    and various tools (e.g. epsilon/3 proofs, passing to a subsequence, etc.), and if you accomplish this without metric space ideas, then incorporating a more general approach making use of metric spaces will be straightforward. Texts to consider looking at: Abbott (introductory), Pugh (less introductory -- could be a sequel to Abbott if you have a lot of time -- but not quite U.S. graduate level), (continued) – Dave L. Renfro Jul 31 '23 at 13:11
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    Goldberg (a standard text used in the 1960s and 1970s that is not so well known now), a few lesser known texts discussed. Possibly of tangential interest is When did US mathematics programs start failing to prepare incoming students for books like "Baby" Rudin? (see also my comments there). – Dave L. Renfro Jul 31 '23 at 13:11
  • @DaveL.Renfro Ok thanks. I mean I I'm a theoretical physics student so I only really have the next couple of months to learn the bulk of analysis before I return to university so I am just trying to get as much done as possible, and as thoroughly as possible. It is my intention at some point next summer to touch on some basic functional analysis. I will have a look at Abbot though as this text is also often praised. – Maximus Jul 31 '23 at 13:46
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    I'm a theoretical physics student --- Likely to be useful (see question, answers, comments): Real analysis for a non-mathematician (actually, I've already cited this) AND Lesson plan to self-teach real analysis to student with comp-sci background. – Dave L. Renfro Jul 31 '23 at 14:19

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As an ancient theacher I suggest: take real line as given, and look to approximation and limits. Ask yourself:"which problem are I solving with such machinery".Then pass from the line to the plane, end so on. Ask yourself "why functional analysis" and go on. Do't be hurry: it takes several years. Bye

lib
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