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I am planning to self-study analysis (understood in the European sense, which I think includes calculus).

I have boiled down my book search to three intro books before I move on to further study. To be honest, I am tempted to read all of them:

Spivak's Calculus

Ross' Elementary Calculus

Abbott's Understanding Analysis.

But in which order? Spivak, despite its title, covers pretty much the same ground, albeit in in 500 as opposed to the 300 pages that the other two need. All of them have many exercises and solutions available, which is why I have selected them. What is important to me, that I learn to write good proofs that also give a glimpse of the reasoning behind. The calculus I learned was rather poor and I have forgotten most of it, so I really would like to learn to prove the basics such as limits, convergence, continuity and so on.

Which one to begin with? As time is limited, would reading one or two suffice? If so, which one?

All the best!

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  • $\begingroup$ If you're in a hurry, study Spivak and Baby Rudin. If you become somewhat suicidal, take a break from Rudin but don't forget to come back to it. $\endgroup$ – lzralbu Jun 2 at 18:55
  • $\begingroup$ @Izralbu Rudin is too advanced. As for Spivak, that book is 200 pages longer than its alternatives, so I don't think I will save time by working through it. My fear is that by foregoing Spivak, I miss some fundamental basics (both in terms of theory and practical skills). I do not want to become like so many other students that know their way around some standard stuff but then fail to prove other similarly standard stuff just because it was not covered in their lecture. I havee seen very good students here in Germany completely incapable of formalising statements into predicate logic. $\endgroup$ – Maximilian Jun 2 at 19:19
  • $\begingroup$ @Maximilian Spivak's Calculus is one of my favourite math books! the exposition is extremely clear, and the problems are the best part of the book; he sometimes makes you prove useful restatements of certain proofs/definitions and introduces new concepts. Even two years after going through the book, I still find myself referring back to it from time to time, and often, and even now, I learn something new everytime I read it. (I've read Rudin and other analysis books, but for a first introduction to calculus/basic analysis, Spivak is the only way to go IMO) $\endgroup$ – peek-a-boo Jun 3 at 4:09
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As far as I can tell, you've covered the basics of calculus insofar as you've done the exercises of 'determine the limit' or 'calculate the integral' and it all seems familiar. So my assumption is that you want to understand why this makes sense, in a rigorous sense.

Abbott is great for pedagogical reasons, and for your intentions it is perfect. I am inclined to say that you do not need the other two, as the attitude in Calculus and Analysis are different; You want rigour, not necessarily the results. Abbott will cover the results in calculus but in the rigorous approach of how the mathematician understands limits; For example, Abbott emphasizes the attitude that integrals are limits of sums rather than antiderivatives. Calculus will not make you understand proofs, but Analysis will.

For additional depth I'll recommend Rudin's Principles of Mathematical Analysis, as it is written in a 'very mathematical way'. I'll emphasize though that understanding analysis is difficult without having done the calculus.

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  • $\begingroup$ Thx. As I have mentioned above I strive for both a deeper understanding of theory and acquisition of skills. By that I mean different ways of proving things and not being restricted in my method of proofs because I have come across only one way of proving things. I want to be as fluent as possible, able to cenceptualise/express myself/solve things in as many ways as possible. At the same time, time is limited, so there has to be a trade-off. $\endgroup$ – Maximilian Jun 2 at 19:25

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