Requesting review of short proof by professor I'm a student enrolled in an university in Italy. I'm currently taking a course on probability theory, but the way is it taught is, whilst very good, not very thorough; I mean that in class some of the most difficult part are skipped. As such, I'm studying those part on the book by myself. Since this is a completely new topic for me, I often wonder if I'm understanding the proofs correctly, and if I'm able to correctly apply the theorems.

For this reason I sent an email to my professor, kindly asking to review a short proof (that I've written in latex, so it was pretty and all) specifying that I wasn't sure if all the operations and the theorems employed were indeed well used. The professor did not answer my mail.

My question is, was I wrong with sending the email in the first place? Is it bad etiquette to do so? I especially don't want to appear like some sort of "smart aleck" or anything, I am honestly trying to understand the topic. Of course I don't mind that the professor didn't reply, I know he must be pretty busy. On the other hand, I also have the feeling that if I can't get any feedback I may as well study the books by myself. What is even the point of being enrolled in the university then?

I hope someone with experience (maybe a professor!) can share their thoughts.
As @paulgarrett has said, there's nothing wrong with sending this sort of email, and (one hopes) many professors will be pleased that a student is showing more initiative than the average.

However, from the professor's perspective: unless the answer is "Yes, that's exactly correct", or "No, you clearly haven't understood the first thing about this course", providing a helpful answer can be quite time-consuming. Explaining why your proof is not quite right may require material that's outside the scope of your current course; it may also require the professor to sit down and produce a lengthy response. Sometimes, there are also nuances of "it's not exactly wrong, but it's not entirely right either" that are difficult to translate into written form. 

Thus, it's all too easy to put this sort of question to one side, with good intentions of replying later that never quite come to fruition. I suggest the best strategy, assuming you've given the professor a reasonable amount of time to respond, is to go see the professor in person (perhaps during his office hours, or after a lecture), and (politely) ask if he's had a chance to look at the proof you sent. Take a print-out with you. You will at least get a better sense of whether this professor welcomes your questions or not.