TITLE: The level of knowledge needed for graduate study
QUESTION [26 upvotes]: I was in class, and I saw a grad student reel off a proof of a theorem in real analysis, one I found quite hard and never really tried to understand thoroughly, with no effort.
This led me to ask the question: at what level should I know the material I learn in my math classes?
In particular, if I want to go to graduate school, what standard should I hold myself up to? Should I be able to reproduce proofs of all of the main theorems from scratch on a whiteboard, completely understanding the all of the techniques used? 
This is not really a standard enforced by the coursework, which requires knowledge of the statements of the theorems and some problem solving ability.
So the question really is: should I come out of undergrad understanding not just the statements of the results in class, but also the proofs?
Also, from personal experience, does taking this approach to studying help one solve mathematical problems?

REPLY [7 votes]: As a first pass answer, what grade do you think you'll be receiving in the course you mentioned? If you're on track for an A, then presumably the professor thinks you are learning the material at a required level. Similarly, a graduate school would look at the grades you have received as one indicator of your ability to graduate-level mathematics. 
As another consideration, you are probably equally fantastical to a Calc I student---How did you take the derivative that fast?! How did you possibly memorize the Mean Value Theorem?! The graduate student you mention has had the benefit of studying this material for much longer and has been better able internalize theorems and proofs that seem dense to you. 
Beating the example of the Mean Value Theorem into the ground using the lens of Terry Tao's excellent article, a progression of understanding might go like: 

Calc I student: No idea what this means, it's a "totally obvious fact." Use it to get the answer in the back of the text, maybe some analogies with speedometers or something?
First analysis course student: Can produce an $\epsilon$ - $\delta$ proof given the exact statement, can (laboriously) construct counter-examples when various hypotheses are removed. 
Grad student: Something about average rate of change and derivatives; remembers that it's a consequence of Rolle's Theorem and has a relationship to Lipschitz continuity. Draw a convincing picture and reverse engineer the hypotheses. Turn the convincing picture into a rigorous proof if desired. 

As a closing thought, memorizing difficult proofs verbatim is a terrible use of time. Instead, study them to learn how the proof is structured, why each of the hypotheses is necessary, and how the theorem fits into the general mathematical landscape. For my own studies, I would rarely memorize exact statement of a  theorem and proof. Instead, I focus on learning what makes that theorem tick---what other theorems/lemmas/definitions does it rely on? What other theorems depend on it? What's the structure of the "standard" proof---induction / contradiction / split into cases / ... ? Can I build simple counter-examples if any of the hypotheses are removed?