So as I have said before in a previous question, I am taking a first course in Mathematical Analysis, and I'm quite excited. I just found out though that unlike the other professors at my university, my professor is using Real Mathematical Analysis by Pugh. I thought it was rather strange because I have read from so many places that Rudin's text on the topic is "the bible" of mathematical analysis, and also he is the only professor who doesn't use it. So I was wondering what some of you experienced mathemeticians thought of choosing this book over Rudin? Is this book a little easier to use than Rudin's? I have heard the Rudin is quite rigorous.

17In my own experience I have seen that the "bible" doesn't exist in mathematics. One needs to know a lot of different books, because every one of them has its strengths and its weaknesses. Even Rudin's "Principles of mathematical analysis" has weaknesses, IMHO. – Giuseppe Negro Dec 07 '12 at 00:47

@GiuseppeNegro What do you think some of the weaknesses of Rudin is? – TheHopefulActuary Dec 07 '12 at 00:50

1There are comments on your question on Amazon (read the user reviews) including some comparative statements to Rudin. See: http://www.amazon.com/RealMathematicalAnalysisCharlesChapman/productreviews/0387952977/ref=dp_top_cm_cr_acr_txt?ie=UTF8&showViewpoints=1 – Amzoti Dec 07 '12 at 00:52

20It is way too concise, to say one. This leads to a tendency to present only the final product, hiding all the reasoning that leads you there. This can be very useful if you are reviewing things you already know, but can be very frustrating if you are studying for the first time. For an example of what I mean try having a look at the first chapter, and precisely at the theorem asserting that every positive real number has a square root. I find that the proof of that theorem is unreadable *unless* you already know what to do. – Giuseppe Negro Dec 07 '12 at 00:58
3 Answers
Every great book on a particular topic will have things another great book omits. Having seen both Rudin and Pugh, I would say they are both excellent choices for a rigorous course in mathematical analysis. Pugh's book may be easier to understand as Rudin is very terse.
Often what we call the bible is just what has been used for many years by many people, but newer alternatives exist and can be equally awesome too.
Seconding Giuseppe Negro's comment that there are no "bibles" in mathematics: what is true is that there are some sources that do accomplish a certain difficult goal, without too many bad side effects, and earn a place in the "pantheon" for this attribute. Sometimes, however, a place is earned for "being impressive" rather than for "being helpful". Or for "being tough" rather than "being clear". One person's "rigor" is another's "tediousness", etc.
Pugh is a genuine mathematician, so the choices he made in putting together his book are surely reasonable. I have the impression that he chose to emphasize intuitive/pictorial things, rather than CauchyWeierstrassian, as would be Rudin's wont.
Most days, I agree (with D'Alembert, I think it was?, who said) "After belief, proof will follow".
EDIT: in fact, as Michael Harris observed, it is "Allez en avant, et la foi vous viendra": "Go forward, and faith will follow"... conceivably even more radical? Or is it less so...? :)
Although years ago I scoffed at this potentially seemingly frivolous unrigorous remark, by now I understand it in a different way. E.g., if one has a "physical intuition" that a thing is true, this often suggests "proof". And, from the other side, if a given question is "purely formal", sometimes meaning "of no genuine interest to anyone", then the "formal" (in a derisive sense) approach we find ourselves taking is indicative of lackofsense.
My advice would be to look at many sources, encompassing a range of viewpoints. One can argue that some sources are too fussy over small things, and that others are negligent. In the end, I think a professional mathematician wants to have experienced a certain amount of fussingoverdetails, but in as many cases as possible seeing, in hindsight, that many of those details were effectively foreordained to be ok, ... rather than conceding that "the universe is hostile, and things tend to be false rather than true..."
That is, while many naive notions are, of course, incorrect, I claim that the good news (about elementary analysis, as of many other things) is that things turn out pretty well. That is, although it is entirely reasonable, and perhaps intellectually responsible, to be worried about details, it turns out that things are not as bad as they might have been. (One may argue that if this were not so we wouldn't be doing this at all.)
One minorbutimportant disclaimer is that essentially all "introductory analysis" sources do limit their technical outlook, so that some questions which can be asked in relatively elementary terms, but which admit no real coherent answer in the same terms, are ... nevertheless... answered in sometimesghastly terms. My own pet case is about differentiation in a parameter inside an integral... :)
Summary: multiple sources. Look around. (And... there are no rules.)
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Interesting quote. I like it, and also happen to agree. Have any source on it? A quick Google turns up zilch. – Noldorin Apr 06 '14 at 00:06

@Noldorin, hm, ... I'd wager that this is quoted in E.T. Bell's (not entirely reliable... not to mention biasladen) "Men of Mathematics", but I don't have my copy here. I can check this coming Monday. – paul garrett Apr 06 '14 at 03:53

Ah right, interesting. Well, please do check if you wouldn't mind; I'd appreciate it. – Noldorin Apr 07 '14 at 00:18

1@Noldorin, ... (found this thing again... :) The (corrected) attribution does not seem to be in Bell's book. but is given on http://turnbull.dcs.stand.ac.uk/~history/Day_files/Day1029.html – paul garrett Apr 07 '14 at 18:02

That's great, thank you. I'm not sure d'Alembert meant precisely the same thing you did in your above answer, but I do think both are very worthwhile pieces of advice! – Noldorin Apr 07 '14 at 18:29

As I generally agree with you but the first chapter of Pugh's book is really confusing; although he has tried to be friendly and understandable. – Hosein Rahnama Oct 08 '17 at 12:47
There are flaws s Rudin book. Taylors theorem is not handled nicely. differential forms are rather messy. implicist inverse not covered very nicely. i would say pugh offers lot of insights. but the exercies are quite hard and measure theory is not good. stokes theorem is not complete. implict and inverse function theorem are good but he should offered general proof in final paragraph which he has left to exercises dieuodennes foundations is classic Bartles elments of realanalysis is a very good bookand easier bt implict inverse are not well handled. langs ntroduces even distributions.book on undergraduate analysis is good and some insights are givem . some thorems are proved in two different ways at different places and comments that interchange order of integration and differentiation under integral sign are corelated. treatment of differential forms is excellent.
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