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Lately, I have been reading a really dense real analysis textbook and I came across different theorems which have exceedingly long proofs (for instance, the dominated/bounded convergence theorem or many other theorems that involve interchanging the order of integration/differentiation).
I wonder if it is worth learning such proofs, because, for instance, I have known the DCT for a while and I have used it extensively to compute different limits, but I can't see why it would be useful to know how to prove it. I would like to add that I am mostly self-learnt at this level, because I have just finished high school. As a result, I don't know if, for instance, in a college level real analysis course the lecturer would prove such a result.
I chose the DCT as an example because I have found it quite useful in different problems, but there are many other theorems which fit into this category.

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    $\begingroup$ You can always come back to it. Textbooks are rarely meant to be read from cover to cover linearly. Once you're comfortable with using a difficult theorem, you can move on and later come back to read the proof. Chances are, you'll have improved in many ways along the way and will find it easier to digest the proof and benefit from it. $\endgroup$
    – Favst
    Sep 17, 2020 at 22:16

3 Answers 3

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While some memorization is inevitable, the most powerful technique in your arsenal should be the ability to recall the key ideas that go into theorem proofs:

  1. Key Theorems that are needed for proof.

  2. Key Counterexamples that will help you remember theorem assumptions.

For example, DCT is a direct consequence of Fatou's Lemma, which itself is a direct consequence of the Monotone Convergence Theorem. In fact, just having such hints will often allow you to reproduce, or at least sketch out the full proof. You can think of this as a form of knowledge compression. In addition to the key ideas, if you remember the assumptions of the theorem, then knowing examples for which it breaks will further reinforce your understanding and recall of their proofs. This is circular, in that if you're suddenly unsure about a theorem's assumptions, a good counterexample can help you remember (do I really need g to be integrable in $|f_n|\leq g$ in DCT?).

For further examples, the monotone convergence theorem requires an increasing sequence of functions, which easily implies a limit always exists (even if its infinite). For the monotone convergence theorem, a key related result is the monotonicity of measures $\lim_n \mu(A_n)=\mu(\lim_n A_n)$ for increasing families of sets $A_n$. For Fatou's lemma, there's a trivial negative counterexample.

This process may seem difficult at first, but I promise that after a bit of time, it will actually make future learning easier because you will see ideas constantly reused, and your knowledge will form a densely connected web, which can self-correct itself when you're fuzzy about one of its nodes. Most importantly, you will have an easier time quickly figuring out which ideas in a theorem are important and which ones can be left to be looked up in times of need.

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Yes. It is worth learning such proofs, however, it is not worth memorizing such proofs. The main idea behind introductory Real Analysis courses/text is to prepare you for the rigor of future mathematics courses/texts. Moreover, rarely will it be important to recall from memory the proof of the Dominated Convergence Theorem, however, training yourself to understand and read such complex/convoluted proofs will turn out to benefit you in the long run.

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    $\begingroup$ Thank you for saying so neatly what I have been trying to summarise as a comment. The important thing, in my view, is making sure you understand the main ideas behind a proof so that you can use them in proofs of your own. A remarkable fact in mathematics is that proofs are often more generally applicable than the theorems that they prove. $\endgroup$
    – Rob Arthan
    Sep 17, 2020 at 22:30
  • $\begingroup$ It is not always worth learning stuff. For example do you really want to read and learn tens of thousands of pages, if all you want is to use the classification of finite simple groups? Sometimes life is just too short. $\endgroup$
    – freakish
    Sep 18, 2020 at 6:51
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"Analysis" literally means "breaking something complex into simple pieces". Attacking anything too complicated directly will usually just give you a bruise from running into a wall. That applies at scales from big open problems, to reading a paper, to textbook exercises. The solution is almost always to just steadily pick away at the edges. There's a famous Grothendieck quote about this (McLarty's translation):

I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

If you don't think they're getting much from reading "highly complicated proofs", it probably means you haven't broken things into enough simple pieces. Trying to pick out the intuition behind the proof is one approach. Moving on to exercises and coming back to the proof later when you're stuck is another.

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