Are difficult exercises good for beginners?
I'm self-studying Rudin's Real and Complex Analysis. I've finished the first two chapters so far, and I don't have any major problems understanding the definitions and theorems. I can prove the theorems on my own. Exercises in chapter 1 were OK, but I'm finding the exercises in chapter 2 to be very difficult. To give you an example, one exercise expects you to come up with the generalized Cantor set on your own. Another is a proof that was published in a journal.
Are such exercises the best way to learn for a beginner? Or is it better to start with a simpler set of exercises that test your understanding of the material before you venture into more difficult things? Should I augment my study with another book that has easier exercises?
I'm feeling frustrated and would like some guidance here. Thank you.
Solution 1:
I think difficult exercises are essentially bad in general: they tend to discourage the student rather than check the understanding of the material.
It so happens that I too tried to read that book by Rudin in my fourth university year.
I found the reading very hard going and there were many exercises I couldn't do: this affected my morale very negatively.
Retrospectively, I find this book dreadful pedagogically and the worst offence is that there are no pictures : this is a mortal sin in a book on a geometric subject like complex analysis.
(In fairness I should add that I do use it as a reference now: it contains sophisticated beautiful results like the theorems of Müntz-Szasz and Mergelyan, which are not often proved in books on holomorphic functions.)
On a more constructive note, let me mention two great books you might consult:
$\bullet $ Lang's Real and Functional Analysis which contains an astonishing wealth in material (including the Haar integral and Schwartz's distributions)
$\bullet \bullet $ Remmert's Theory of Complex Functions , written by a genuine master and containing, apart from a perfect technical treatment, invaluable historical vignettes.
Finally, for exercises proper, an excellent source is Schaum's Outlines series .
The books there are very user-friendly and the exercises quite reasonable, with a progression from very easy to more demanding, accompanied by clear, detailed solutions.
Look here for the dirt cheap volume ($13.41 !) on Complex Variables.