Is it important to be able to remember all proofs in my course? I am following the course materials for module Pure mathematics from Open university. It includes a significant number of proofs. While I mostly understand the proofs. I forget the details about 5 minutes after reading a proof. My question is this: Is it important, when somebody says: "Monotone convergence theorem", to be able to rattle off the proof? I mean should I know all proofs by heart? Importantly, I am not taking the course, so I am not pressured by grades, but I do want to be able to say that I´ve mastered the material. Any advice?
 A: Do not try to memorize the whole proof. After reading a proof, write on paper a summary of it which is 1/4 the length of the original. When you have done this to all the
proofs on your course, reread the summaries and make sure they remind you of the original proof.  Then summarize the summaries.
A: I still have a long way until becoming a professional mathematician. That said, this is something that also used to worry me and, after years of undergraduate math, what I’ve gathered is the following set of guidelines.

*

*Forgetting stuff is inevitable! After learning, say, Complex Analysis, if you don’t use it for a while, most of the stuff is going to be a bit fuzzy - it’s normal!

*The ideas of the proofs and theorems, however, should be what you try to absorb. You cite the Monotone Convergence Theorem. Instead of remembering the precise $\varepsilon-\delta$ proof, try to just reason “well, if a sequence is growing, never coming back, but it has a bound, at some point it has to stop” and build an argument from that.

*Another part that is very useful to remember are some references. Maybe, once you hear about the Monotone Convergence Theorem in the future, it won’t be a complete stranger. You’ll think “I’ve heard that somewhere… was it in real analysis?” At that point, you should remember the name of some real analysis book to look it up and refresh yourself.

For these three points, @Yuval’s method can help, and writing things down is in general extremely helpful.
In short, no, you aren’t expected to always recall $100\%$ of proofs, or even theorems. The main points are the ideas behind them, the names of the major ones, and places where you can find more details should you need them.
Hope it helps!
