Requesting review of short proof by professor
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.