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?