@CJ2116: I just took a look at that part of Abbott's real analysis book that is visible on amazon and compared it to baby Rudin. Abbott was at first much too verbose and chatty for my taste, but when I located and read some of the definitions and proofs of theorems, such as countability of the rationals, and uncountability of the reals, I found it equally as rigorous as Rudin, but far more clear and user friendly, while giving essentially the same arguments. So I do not consider Abbott much of a come - down at all in terms of rigor, just a vast enhancement in clarity.
Both of them e.g. define a function as a vague "rule" or "association" between elements of sets, neither one making it entirely precise by introducing the notion of the graph. The nice proof of uncountability of the reals via the shrinking intervals property of the reals, is one I had not seen before, and is in both. I thought the explanation of it in Abbott was much clearer and equally rigorous as the same argument in Rudin. Both arguments for the countability of the rationals seemed about equally, and not totally, rigorous in both, but again clearer in Abbott. The nice avoidance of the problem of repeated occurrences of the same rational in the set of pairs of integers was beautifully dealt with in Abbott, if slightly colloquially, and also in essentially the same way in Rudin, with a bit less clarity, but still without complete precision in my opinion. (In Rudin the slightly incomplete part is in the proof of the result 2.8 that an infinite subset of a countable set is also countable, where he asserts but does not prove that his construction gives a bijection. In Abbott the slightly colloquial part is where he argues that a countable union of finite sets is countable by just displaying them side by side, in order of index, rather than giving an inductive definition of the enumeration. Some may prefer Rudin's more precise inductive definition of his bijection, but since he omits the proof that it actually is a bijection, he seems to become guilty of the same incompleteness of rigor. ... Aha, I have read further in Abbott and found exactly the same inductive definition for the same lemma of Rudin, as an exercise in Abbott. So what Rudin omits, Abbott gives explicitly as an exercise. So which is more rigorous, omitting an argument without comment, or assigning it as an exercise? And neither of them even state the crucial well ordering property of the integers, although both use it in this proof.)
OK here is a comparison for you to assess the relative clarity and ease of reading of essentially the same proof in both books: compare the proof of Thm. 2.43 in Rudin, with Th. 1.56, pp 28-9 of Abbott.
I have not read anywhere near all of Abbott of course and may have gotten my favorable impression from only a small, possibly unrepresentative, sample. It may also be that Abbott's arguments look rigorous to me because I know how to fill in the details. I still feel, even as a retired senior mathematician, that Rudin is not a welcome source for either learning or even summarizing the material. For me, with over 40 years teaching and research experience past my PhD, I thought that by now Rudin would indeed read like a useful quick summary of the facts, but even today it was still less readable even for me, than Abbott. In particular, a summary should be clear. I have heard from people who like Rudin, but I never recommend it to anyone for learning analysis.
To be fair, I have written a book with similar failings myself, on linear algebra, where my goal was to see how briefly I could cover the material. I managed to provide more theoretical "coverage" than is in many introductory linear algebra books, and in only 15 pages, and posted links to it here. (I later expanded it to 120 pages, still short by usual standards.) I think Rudin also was trying to see just how succinctly he could summarize the material, whereas Abbott was trying to teach it. Abbott went too far, for me, in his voluble writing style, but students seem to like it, in contrast to Rudin.
My point is, even if Abbott's proofs look less
formal than Rudin's, in my opinion they are reasonably rigorous, and I suspect you will learn more from it than you did from Rudin. One difference is the coverage is greater in Rudin. I.e. in a bit over 300 pages, Abbott covers only the material in the first 2/3 of Rudins 330 pages. But for that one variable material, I recommend Abbott.
Please forgive me for always jumping into every discussion of Rudin on the "against" side. I had to teach out of that thing, maybe that is the source of my frustration. But I like Dieudonne', which is even more condensed, but I think much better and more insightful. I.e. I learned things from it, (even if my class may not have).
One more remark, this time in Rudin's favor. The fact that Rudin has been around for so long, and his arguments are solidly rigorous, means that later writers can borrow from his material in substance, and work at improving the presentation. So in a way, many later analysis books, may be something of reworkings of Rudin, or if not, at least they had Rudin available for consultation if they chose. Of course Abbott says explicitly that it is the book of Bartle to which he owes his own education. I am not as familiar with that one. The books appeared in roughly 1964 (Rudin), 1983 (Bartle), and 2001 (Abbott), so each of the last two authors at least had the option to have access to Rudin.
One reason I realized this was the fact that Abbott has the same argument, via intersections of shrinking compact sets, for uncountability of the reals, as Rudin does, and I thought I had not seen it before. Actually I realize after thinking about it, it is basically the exact same idea as Cantor's diagonal argument that I had seen in high school, where he constructs a decimal that is not anyone of those in a given list. Namely you determine a real number by a sequence of things, and you arrange successively each subsequent one of those things so as to rule out the nth listed number, eventually ruling them all out from your list. It does not matter whether the sequence is a sequence of decimals digits or a sequence of compact intervals. In fact a sequence of decimals can obviously be viewed as a sequence of intervals, with the nth one of length 10^(-n). I.e. the decimal number π = 3.141592653589793..., is the intersection of the intervals [3,4], [3.1,3.2], [3.14, 3.15],... So even though I hadn't seen it phrased this way, there is no new idea other than Cantor's original one. Thus it was not necessary to have read Rudin to come up with this argument, which is probably very old.
