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I am currently reading Abbott's Understanding Analysis and also trying to solve the problems. However, I get stuck on some problems and wonder why is that the case because people say that Abbott's book is just introductory.

I know that getting stuck for a while is a part of learning real analysis but how should one go further? Should I aim at solving all the exercises before moving on to the next section or chapter? Or should I solve as many as I can within a certain time frame and move on to next sections? (Of course I'll get back to the remaining exercises and try to solve them) I do not read the latter sections unless and until I feel like I have understood the preceding ones.

I'd like to know how you progressed in the process of learning analysis. Thanks :)

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    $\begingroup$ It takes time to get ‘used’ to the new ideas. So it is ok to struggle a bit. You probably will not ‘understand’ everything immediately but you can come back to it. Ask for help when you need it. $\endgroup$
    – AnyAD
    Jul 9, 2020 at 8:07
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    $\begingroup$ If you understand the main body of the text well, including the proofs, then the book is probably at the right level for you. This is how I would approach things at this stage of learning. If I worked on a problem for maybe an hour and didn't make significant progress, I would usually stop and come back to it later. It would go on my list of unsolved problems. Sometimes I'd solve it later the same day, sometimes days, weeks or months later, and on a few occasions never. But I revisited my unsolved problems... $\endgroup$
    – Anonymous
    Jul 9, 2020 at 10:16
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    $\begingroup$ regularly. There was no answer book and no internet, so I couldn't just look up an answer. That being said, I rarely found that my inability to solve a problem reflected a serious lack of understanding of a chapter. If you find there are a large proportion of the problems that you're not solving, it might be good to look up a solution (or ask for one) within a matter of days or weeks, depending on your patience. Generally, I felt comfortable moving on to the next chapter when I'd made a good effort on all the problems. $\endgroup$
    – Anonymous
    Jul 9, 2020 at 10:19

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There's really no royal path to learning Mathematics. Understanding Analysis (eyy puns for days) can be difficult and does require quite a bit of work. There are quite a few things you should consider:

  1. Is the book you're using the right one for you? Do you enjoy reading it?

You know, for me, I would not use a book that I didn't enjoy reading. If you're not enjoying reading that book, perhaps other books would be more suitable for you. There are many other Analysis textbooks available, including ones that have been translated from other languages into English. Have a look at them and see which one resonates with you as a person!

For instance, do you want to know what my first introduction to Linear Algebra was? It was the Linear Algebra book by Klaus Janich. That was my first time dabbling with proofs and the book was hard to work through, mostly because the author was very minimalistic with the words he used to describe something. So, very little fluff but also a lot to figure out.

But you know what? I loved his writing style and I loved the memes. So, I continued reading it :D. Really, read what gives you enjoyment.

  1. Do you have the correct background? A lot of people say that you need Calculus for Analysis. But what kind of Calculus have you learnt, if you have learnt it at all? Is it the stuff from cookie-cutter textbooks or is it from books by Richard Silverman, Michael Spivak or by Richard Courant?

In my mind, I think that the correct background for Analysis revolves around a rigorous treatment of calculus. You know, a rigorous treatment that you'd get from Spivak's book or Silverman's textbook or Courant's textbooks.

A large portion of what's usually covered in Analysis would be discussed in there but just enough would be left out so that you wouldn't get too caught up in the extreme details. The extreme details are what you'd discuss in an Analysis class.

These books would also teach you how to do proofs in a way that's more gentle than the treatment by Analysis textbooks. That's not to say that you can't use an Analysis textbook to introduce yourself to proofs. But it'll be a bit of a difficult climb.

  1. Are you studying it in the correct way? Are you simply reading through the material or are you doing the problems? Are you taking notes?

Remember, proving theorems isn't always an easy thing to do. It certainly can be very difficult in many cases. Do you read with a pen? Have you considered trying to prove all of the theorems in the text on your own first?

There's an amazing community over here that's willing to assist you in understanding what you're doing. Make use of that and come forth with your solutions!

One thing that I've learned is that you should learn to get rekt by your problems and have others tear your proposed solutions apart. That's how you'll learn best.

If you have any further questions, don't hesitate to ask. I think that what I've said above adequately describes the way that I approach Mathematics as a whole so if you want any other details, I'll be happy to provide them.

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    $\begingroup$ @ak47: Well no one really expects you to solve all problems of one chapter before moving to next chapter. By design some of the problems (a minor portion) are challenging and may need more thinking and effort, but in general you should be able to solve other problems. If there is an issue with majority of problems then it means you need to read the chapter again. $\endgroup$
    – Paramanand Singh
    Jul 9, 2020 at 15:50
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    $\begingroup$ +1 for the points you have raised. And seriously one should not read books which one doesn't enjoy (unless insisted by your teacher). $\endgroup$
    – Paramanand Singh
    Jul 9, 2020 at 15:54
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    $\begingroup$ Yeap, too many people like to point out how different reading a novel is to reading a math textbook. They get so lost in the symbols and the theorems that they forget that they actually should be enjoying this stuff more than anything. $\endgroup$
    – Abhi
    Jul 9, 2020 at 16:44
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    $\begingroup$ @ak47 While it is true that the inability to solve a problem rarely reflects lack of understanding, I would say that the inability to START trying to solve a problem is an accurate reflection of the lack of understanding. You see, if a problem is difficult, then that's fine. Everyone struggles at some point. Ultimately, though, you must do some reflection. Are you struggling because you can't quite find a way to continue with what you have? Or are you struggling simply because you have no way to start? $\endgroup$
    – Abhi
    Jul 9, 2020 at 16:46
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    $\begingroup$ In a sense, I feel that too many people view Mathematics as something so exact and rigorous that one should go through their Analysis course proving theorem after theorem. You have to remember that Mathematics was done by people and it's a story of the struggle of humanity to come to terms with the limits of our imagination. You must not only do Mathematics. You must constantly reflect on it. WHY aren't you able to understand what's going on? WHY was the idea EVER explored in the first place? WHY do people care if $\lim_{x \to 0} \frac{\sin(x)\}{x} = 1$? There's a human part to all of this. $\endgroup$
    – Abhi
    Jul 9, 2020 at 16:48

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