Most of the content is new to me and there are a lot of theorems and proofs that I am learning; not that I need to know all of them but I enjoy to learn more. Some of the concepts (like open sets) or some of the proofs are quite time consuming; for instance, it takes me about 45 minutes to understand a hard proof. Is this normal?
[Math] How to study for hard math proofs
proof-writingself-learningsoft-question
Best Answer
Regarding remembering lengthy proofs, my technique is always to take that proof through a certain process:
It does require work, but there is no shortcut. This is not as time consuming as it may sound compared to the outcome.
As others have mentioned, do not go into this much detail with every tiny proof you stumble upon. You have to consider each time if it is relevant and serves a goal.
Regarding reconstructing proofs at a much later date, I must say that the mind works in mysterious ways. Over time you internalize techniques and even theorems on a higher level, so that these become readily available for tackling other problems. That has already happened to you with some of the techniques from previous courses. Otherwise you would not enjoy math so much by now.
I have taught at Danish Upper Secondary Schools for years now, and all those proofs have become so simple now, that I do not have to remember much to reconstruct them. I just apply the techniques I know and my general idea of what needs to be done, and in that manner I can reconstruct pretty much three volumes from memory at any date, I think. But these are simple things like basic proofs in differential calculus and vector calculus and the like.
Finally, just two words about my hero, Leibniz. He thought that the use of well suited language and notation was key to widening your understanding. He invented the notation $dx$ and $\int$ which to some degree shows this trademark. I too like re-phrasing proofs to try and make the language and notation a better vehicle for understanding!