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(I am aware that some people might frown on this question, but I had no place to ask; this will definitely be voted to be closed in SE. I apologize in advance.)

I am currently a 1st year grad student and I would like to ask for your advice on how to study math. I am wondering to what extent do I have to understand proofs in the lecture. I understand what is going on in the lecture, but if someone asks me to regenerate proof, then I cannot do it. To be more specific, if the proof solely consists of a bunch of inequalities, it is unlikely that I can regenerate proof; on the other hand, if the proof relies on one good idea, then I can provide key insight behind the proof even though I cannot provide a complete proof. I keep practicing until I can redo the proof by myself, but this has been too exhaustive and time-consuming.

Also, I forgot a lot of proofs in undergraduate math; for example, I don’t remember how to prove L’hospital’s rule (even though I recall that it uses mean value theorem), and it feels bad to be using the theorem without remembering the proof. Ideally, I would like to remember every proof I do, but I am not smart enough for it. Can anyone provide a tip and share your experience how you handled this?

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    $\begingroup$ Sorry, but this not a very good question for Mathoverflow. Having said that, try to understand the ideas behind the proofs, and not necessarily every detail. $\endgroup$ Commented Apr 6, 2021 at 11:23
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    $\begingroup$ This will also be closed on MO. I'm not sure what advice will be most helpful in your individual case (you didn't specify which specific field you are studying, e.g.), but generally it is best to remember and understand certain general proof strategies instead of every proof of every lemma you encounter in lectures. You will find that many proofs in a given field are broadly similar in what ideas they draw on and what kinds of general steps they follow. $\endgroup$
    – gmvh
    Commented Apr 6, 2021 at 11:26
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    $\begingroup$ On my opinion, the only way to understand mathematics is by doing it, that is by solving problems. $\endgroup$ Commented Apr 6, 2021 at 11:46
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    $\begingroup$ I would agree that you don't have to remember all the proofs (although you should remember some of them) and that doing problems is the most important things. I would add that, if you don't like proofs that have a lot of inequalities, one solution is to specialize in a field of mathematics that has few or no such proofs - generally speaking, a field of algebra rather than a field of analysis. There are certainly skilled researchers in algebraic geometry, number theory, and related fields who are not comfortable with proofs involving many inequalities. $\endgroup$
    – Will Sawin
    Commented Apr 6, 2021 at 13:14
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    $\begingroup$ @Donu Arapura: Well, on that basis it won't be worth while asking research questions on the internet. It ought to be better to ask a trusted advisor - whereas this site is devoted to the former! I think its advisable to allow a certain proportion of soft questions. $\endgroup$ Commented Apr 6, 2021 at 14:13

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Ideally, I would like to remember every proof that I do.

Why? I've made the same mistake.

Not every proof is worth remembering. Sometimes a much better proof comes much later. Usually, it is the strategies used in a proof that are worth remembering. Also it's the understanding of what a whole set of results are building towards that are important. In other words, its not enough to know the details, you also understand the bigger picture. Often it's this bigger picture that enables you to understand the pount of the details.

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