How I deal with lack of exercise in pure math books?

Question

I am trying to teach myself real analysis, but I have a problem. There aren't enough exercises unlike calculus, which has around 100 problems per section (which is too much). Real analysis only has about 10 problems per section (which is too few). I find that when I move from one chapter to another, I forget the proof of certain theorems. Additionally, I don't think I have fully mastered the concepts of real analysis yet that I have read. Is this normal? How do you deal with this problem? (I hope I am not the only one facing it.) Should I use multiple books and solve problems in each one?

Answer 1

"Normal" is quite subjective, since a different learning style and learning trajectory is suited for each person. However, if you're feeling concerned that you're forgetting the proofs and that you haven't mastered, then I would recommend trying to address those concerns by spending more time (such as exercises).

But make sure the concerns are internal for yourself, and not due to your expectation of how other people learn or what other people may expect of you. For example, I have forgotten a lot of the proofs and even theorems that I learnt. And sure I could have done things better if I remembered them better. But I felt they weren't that important to remember, and I think I've done an ok job with my life (I did get a PhD).

As for how to go about your concerns, sure you can try multiple books. But be careful not to burn yourself out doing too much. Set a limit and decide to move on, even if you may not feel perfectly comfortable.

Ultimately, your question is more suited for a professional trainer; though those can be expensive. I personally just experiment trial-and-error; learning from what worked and didn't work for me to keep improving my strategy.

Answer 2

Even if you think you can "fully master" something by solving a lot of exercises, you will still forget the details after a month or two and you will be doubting if you've "mastered" the material well enough back then. That's why reviewing is so important. However, you shouldn't worry too much about it -- you will find it so much easier when you revisit the material, even if you've forgotten the precise definitions/theorem statements/theorem proofs, etc.

Just move along if you think the things you've grasped at the moment is enough for learning the next chapter, and skip the things you find too hard at this point, and review whenever necessary. Of course, you can use multiple books if you find the exposition in your current reference unsatisfying (and I encourage that, since it gives you different perspectives on things); but please don't think of this solely for the purpose of "finding more exercises".

Answer 3

Baby Rudin has about 30 exercises per chapter.

The point Real Analysis is to put a rigorous foundation underneath calculus. However, you already know calculus. This can be frustrating for students as the class seems to be lacking in some direction. "I already know these concepts, why do I need to learn these proofs." But, the proofs are the essence of the class.

Learn the proofs. Don't try to memorize them. But, you should be able to look at just about any theorem and reason your own way through the proof. This may cause you to proceed at what seems like a glacial pace.

Real Analysis may be the first rigorous class that a student has taken. It often serves as a student's introduction to proof-writing. If you are self-studying, it is difficult to gain a sense of whether your proofs are well-written.

Answer 4

I randomly remembered this question and I want to share my experience.

When I first started Real Analysis, it was so boring and I almost lost interest in studying it. This was because 1) most of the theorems I already knew from calculus and I felt that I was just wasting my time with them again just to learn their proofs (I asked about whether Real Analysis has new theorems here) 2) most of the exercises in “Introduction to Real Analysis Book by Robert G. Bartle” were not that challenging and I felt that I would forget the proofs of these theorems if I didn’t have enough exercises so I thought I need more of them.

Then after eight months, what changed? I realized that the beginning of Real Analysis has significant overlap with the curriculum of Calculus courses and after finishing half of “Introduction to Real Analysis Book by Robert G. Bartle” the new theorems started to appear. About the “boring exercises”, I think because Bartle’s book is a very elementary introduction to Real Analysis, it shouldn’t have difficult exercises but for me they were to easy and my motivation to do the exercises was lower after every section to the point that I stopped writing the proofs after chapter 7 and just do them in my head. But it all changed when I read “Baby Rudin”. Not only were the exercises very challenging, but it also gave me motivation to read the book in half the time that it took me to read Bartle’s book and I did almost all the exercises in Rudin.

Conclusion: If someone felt the same as me (bored) because of the lack of new theorems, they will come but after some chapters. And if someone was bored because of the lack of challenging problems, then maybe they should try Rudin’s book.