TITLE: When has one sufficiently mastered an area of mathematics?
QUESTION [34 upvotes]: This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. 
In particular, say, when has one mastered undergraduate analysis? Is it realistic to expect some individual to be able to prove every theorem and do every exercise in Baby Rudin? What about analysis not covered in Baby Rudin? Say, should one also be able to prove compactness is equivalent to sequential compactness, and sequential compactness is equivalent to closed and boundedness in $\mathbb{R}^n$? Should one be able to prove every major statement about completeness? Continuity? Integration?
Or is it enough to be able to read and understand any proof in a particular area?
When can one feel satisfied with what they know about a particular area of basic mathematics? Can the seasoned mathematician prove these theorems by heart without much thought? If so, is this due to brute memorization, or simply mathematical maturity?
In short, what should one reasonably expect an undergraduate to know about any particular subject after completing a course in said subject? I ask specifically about undergraduate subjects, since these are often the areas expected to be mastered by the time one begins graduate school and beyond. This question applies to all areas of mathematics, graduate and undergraduate.  
Interested to hear different interpretations of the words sufficiently and mastered.

REPLY [4 votes]: The deeper we dive into the ocean of mathematics, the stronger is my conviction that we actually do not know anything. Let me give some examples: 
$1.$ Every polynomial $p(z)$ has a root in $\mathbb{C}$
$2.$ For a Hilbert space $H$, $H=W\oplus W^{\perp}$, where $W$ is a closed subspace of $H$ and $W^{\perp}$ is its orthogonal complement.
$3.$ Every subgroup of an abelian group is normal.
$4.$ Compactness is the next best thing to finiteness. 
This is just a representative list. There are so many other powerful results which, when I look at, leaves me with a sense of perplexity. Can this hold ? Is mathematics so very general ?
Further introspection gives me a conviction that all these have already existed. We have found them out late. All the deepest and the most beautiful results come from within. So my firm belief is that, we can never say that one has mastered any branch of mathematics for it is impossible. Whatever we know is just a fragment of that infinite ocean. 
This is just my view. Though there are so many answers already posted, I still felt like answering this question.