Rudin against Pugh for Textbook for First Course in Real Analysis

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 Cauchy-Weierstrassian, 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 lack-of-sense.

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 fussing-over-details, but in as many cases as possible seeing, in hindsight, that many of those details were effectively fore-ordained 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 minor-but-important 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 sometimes-ghastly terms. My own pet case is about differentiation in a parameter inside an integral... :)

Summary: multiple sources. Look around. (And... there are no rules.)