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Please excuse my terrible english skills.

About a year ago, I decided to study basic mathematics. At this time, I was not even able to find a solution of some tricky linear equation.

I studied basic algebra, euclidean and analytic geometry, linear and quadratic, polynomial functions, trigonometry and some basic calculus. (Also stochastics, but we dont have to regard it) These stuffs cover most of the high school mathematics at least in my country. And it should be sufficient knowledge to study so called introductory real analysis.

I never had college calculus. And I am also not willing to spend my time for it. (Stewart, Thomas.. etc.)

Now I am learning university mathematics for myself since few months due to interests in the subject. First I studied naive set theory with the first chapter of Munkres,Topology. Then I jumped directly into Baby rudin. I didn't have any serious problem until chapter 2, basic topology was pretty enjoyable. I could solve most of the exercises. Nevertheless I got stuck in Theorem 3.20 (Some special sequences). I can just barely understand things from this part until now, although I tried and tried.

Is there some easier book than Rudin, which is written for absolute beginners with weak calculus knowledge? And can someone explain me why I can't go through that chapter? I want to learn Real Analysis. Please help me

Summ
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    A slightly easier (but still not easy) book than Rudin is Michael Spivak's Calculus (NOT his Calculus on Manifolds... that's an entirely different beast). THis is an extremely well written book, with few errors, and VERY VERY VERY good problems. Even though I finished the book 2 years ago, I still find myself referring to it from time to time :) – peek-a-boo Mar 18 '20 at 23:14
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    I think you are quite right not to go with those enormous, expensive, thick college Calculus textbooks. They are dreadful ! – Simon Mar 18 '20 at 23:37
  • I quite like Kenneth Ross's "Elementary Analysis: A Theory of Calculus." The book is very well-written, and much, much easier than Rudin, which I personally view as a book that is far better suited to a second course in analysis than a self-study. Spivak's Calculus, as peek-a-boo suggested, is another outstanding book. – John P. Mar 18 '20 at 23:37
  • Does this answer your question? Good book for self study of a First Course in Real Analysis –  May 07 '23 at 22:18

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I really like "Understanding Analysis" by Stephen Abbott:

https://www.springer.com/gp/book/9781493927111

Some say it is not rigorous, but in my opinion it is perfectly rigorous enough. It also motivates the definitions and theorems very well, hence the word "Understanding" in the title.

I also recommend Tao's "Analysis I" and "Analysis II":

https://www.amazon.com/Analysis-Third-Texts-Readings-Mathematics/dp/9380250649

I also really like Carothers' "Real Analysis":

https://www.amazon.com/Real-Analysis-N-L-Carothers/dp/0521497566

All authors really convince the reader that a) They understand the topic in great depth, and b) They want to share their understanding with you. This is surprisingly rare !

Simon
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    You were faster just by a few seconds :-) Definitely my favorite (not only) at the moment - I teach a full course of MA (four semesters) and I use this one for a part of it. I mean Abbott's - I don't like Tao's that much. – Roman Hric Mar 18 '20 at 23:20
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    Isn't it excellent ?! The real men at the beach will kick sand in our faces for going with the wimps' option instead of Rudin, but I tend to think they aren't very happy inside :( – Simon Mar 18 '20 at 23:22
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    Definitely... I never use just one source (some students might not be completely happy about that) but the one by Abbott is a true delight. – Roman Hric Mar 18 '20 at 23:27
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    Didn't he write Flatland? –  Mar 19 '20 at 02:31
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    I think that was Edwin Abbott, much earlier. – Simon Mar 19 '20 at 02:39
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    Some say it is not rigorous --- I'd be interested in a specific example of what some might consider in this book as not rigorous. The reason I ask is that I suspect rigor is being confused with formalism (for example, see the 2nd paragraph of this answer). – Dave L. Renfro Mar 19 '20 at 07:08
  • @Dave Renfro, I recall hearing this opinion (that Abbott's book lacks rigour) in face-to-face conversation, and also reading it here on stackexchange, and I also recall that no-one supported this contention with any examples. Rather it was about hearsay. I suspect that they had not read the book, but were just repeating a mantra. Perhaps I am paranoid ! I think you are absolutely right that some (less experienced) people might make the confusion described in that nice answer to which you gave a link. However, more experienced people... – Simon Mar 19 '20 at 22:34
  • ...(such as the Professor at one of the highly reputed Universities of California with whom I discussed this face-to-face) should know better than to make that mistake. In such cases, I suspect less honourable motives ! In fact, serious question: Should I remove that comment from my answer, since although I completely disagree with those who say Abbott lacks rigour, just by repeating the opinion I risk reinforcing it ? – Simon Mar 19 '20 at 22:35
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    @Simon: No, I don't think you should remove that part, since you're just saying what others have said, and maybe what you wrote (and these comments) will motivate someone who knows more about this issue to elaborate on it. Maybe the comments are partly because Abbott does a lot less with Peano axioms and complete ordered field stuff (careful statements of the various axioms and proofs -- or at least acknowledgement that proofs are logically needed -- of many of the basic properties of real numbers) than Rudin does. I also note in Definition 1.3.2 (p. 14) "the least upper bound" (continued) – Dave L. Renfro Mar 20 '20 at 05:54
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    is used when the uniqueness of the object has not yet been shown (or mentioned). However, looking a little more, I see this issue is addressed at the bottom of p. 14, so this is likely a pedagogical choice. Speaking of rigor, this question goes into aspects that mathematicians often ignore (some on purpose, others without realizing it), so it's not always clear where one draws the line for rigor. Regarding the distinction between formalism and rigor (not all uses of the words, but the specific usage in the answer I cited earlier), – Dave L. Renfro Mar 20 '20 at 06:05
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    this question is a good illustrative example of formalism without much rigor (or at least a lot of symbol clutter without much in the way of being a convincing human-readable proof). See also my comments to this answer and Terence Tao's third (post-rigorous) stage of mathematical education, keeping in mind that you shouldn't jump to the third stage before mastering the first two stages. – Dave L. Renfro Mar 20 '20 at 06:11
  • Very interesting points ! Tao's last paragraph is a timely reminder to me that articles by post-rigorous mathematicians that are littered with "local errors" are not always written with malicious intent ! – Simon Mar 20 '20 at 19:25
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I was also going to suggest Abbott's "Real Analysis" book because it is intuitive and inviting for someone who is having their first exposure to real analysis. It also has some very nice exercises that are great for strengthening one's understanding of the key concepts (it asks to fill in details for proofs given in the book, which one would always do anyway in an ideal world).

Joshua Siktar
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There's "baby Rudin". It's a good book. Principles of Mathematical Analysis series Then when you get more advanced, as the name suggests, there's another book.

There are others, some I was aware of. I think Royden has one. When I started grad school at UCLA, we used Wheeden and Zygmund (or something like that). It had all the monotone, dominated, etc. convergence theorems. We learned about $L_p,l_p$, convolutions, kernels etc. etc.

Oh yeah, and don't forget Folland. He's up at U of Washington. His book is impressive.

Oh, and how could I forget, Rosenlicht. He was an excellent (as usual) Berkeley professor. Pardon my not having read what you wrote more carefully before posting. But I really like Rosenlicht as an intro to the subject. Besides it having sentimental value. It's a beautiful little book, published by Dover.

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user 403337's answer appears unfitting, because OP already wrote that (s)he found Baby Rudin and Rudin too difficult. I disagree with the recommendations of Folland, Baby Rudin, Royden. These are for a second course in Real Analysis, not a beginner.

Stephen Abbott's Understanding Analysis has a solutions manual, but FOR INSTRUCTORS ONLY. You can find it online, but Abbott did not intend it for anyone to download. I dislike how it's wholly black and white.

I love Amol Sasane's The How and Why of One Variable Calculus, particularly its diagrams in color. This title is misnomer — This book covers Real Analysis, not merely one variable calculus.

The exercises are plentiful, well-selected and well-constructed. Detailed solutions to all exercises are provided; they fill the last 150 pages of the book.

Next comes more than one hundred and fifty pages assigned to providing the full solutions to every exercise given in the previous six chapters.