what is expected from a PhD student? As a PhD student in applied mathematics or mathematics in general, are you expected to be able to prove every problem, for example, in an elementary real analysis book? I know it sounds silly but I am wondering if I have high expectations of myself...
For example, some theorems in an elementary real analysis book have proofs which are lengthy and time consuming to understand. Still I try to understand them and manage to do so. But if you ask me about the theorem, say after 2 months, there is a high chance I'd have forgotten how to prove it. I might have ideas but I cannot solve it in totality.
What I'm asking is, I guess, is it normal to understand something in math and forget about it? Or does the fact that you forgot about it/how to do it an indication of not having understood the subject in full in the first place?

When I read a math book, and arrive at a proof I go through the following process:


*

*Try and prove the theorem without looking at the proof. Most of the time I fail.

*Read the proof and write it out by hand. The purpose of this step is to learn techniques which might be used later. 

*Summarize the proof in one or two sentences. 


The summary is what you need to remember because more often than not you can use it to reconstruct the original proof. So to answer your question, yes it is very normal to forget the details in a proof but you should understand the idea of the proof.
