I am in a graduate math program in the US, and the professor is teaching via online lectures. The proof she provideed for one of the theorems is a little bit tricky to understand. I think I found a proof that, though is about the same length, I think is much simpler. Could it be considered inappropriate to email the professor out of the blue and give her the proof?
Email the professor, explain that you came up with an alternate proof, and ask if it is correct.
If you approach it this way, it should be perfectly acceptable to anyone. You get to find out if your more-clear approach is correct, and assuming it is, the professor maybe uses it in the future. If it's not correct, you get to learn where you went wrong. A good exercise for all involved.
It’s okay to ask if your proof is right and, if it is, ask if there is an advantage in presenting an apparently more complicated version of the proof.
Keep in mind that, when teaching, the technique used in solving a problem can be more important than the solution itself. It might be that this technique comes again later in the course, so having seen it once will help the second time around.
Just to mention this possibility: it is very well possible that the professor finds her proof simpler than yours. Different people have different opinions on what is simpler / clearer, when it comes to proofs.
Do send her an e-mail, but maybe it is better not to be too absolute with "this is simpler than the other proof" when you write it.
Jokingly. When I was in my first year there was an exam in calculus. Part of it consists of student preforming a proof of their "favourite" law (one on the list).
Two students were able to reduce three-pages long proof just in three claims proof and one of them, after fair coin toss, walked in and performed this proof.
There were two professors assessing students, because there were too many students in the course so both were taking the lectures, plus one guest.
One of them and the guest were insulted by such mocking of the nice three pages of math (it was way too much shortening for them) and were attacking the student's claims and challenging the reasoning.
The other one enjoyed the proof and the discussion it led to.
In the end the student got mark A and just after their exam the professor who enjoyed the time crossed out the proof from the list for ever and since then they use the proof in their lectures and always naming it as Merryman-Trueman proof, after the two students.
A good professor will examine your reasoning and discuss it with you. Good professor will think high of your interest in th topic and your abbility to think outside of the box instead of memorizing-the-poem.
Bad professor will feel attacked by such arrogance of yours to undermine their genius...
It also depends how you word your email but in any case You will find what type of the professor you have encountered. Just stay polite.
Speaking as an erstwhile adjunct professor, my answer is:
I not only don't see anything wrong with that, I would encourage it. Obviously, you want to be polite about it, but otherwise, I think it's a valuable learning experience.
Of course, it could be that the prof knew of your easier proof but did not use it, for didactic reasons. Or it could be that the prof didn't know of it, or it could be that your proof has a mistake, or that it isn't as general as the proof offered.
I (in the U.S., in math...) would say "of course it's ok to send your own proofs to your professors for comment".
Asserting that your proof is "better", or similar stuff, is probably not productive. For that matter, as in other answers and comments, one of the most profitable outcomes is that you receive a professional critique of your argument, from your professor.
It's not that older people have special powers... apart from having lots of experience, which is almost like a special power, if they've been paying attention. :)
I can only second Jeff's suggestion. Email the prof, say that you found an alternative proof and ask if it is correct. It is then up to the prof to decide whether or not it is "simpler".
I give my students props from contacting me with suggestions and if you play it that way I cannot see how this would be wrong or inappropriate. The worst case scenario is that either your proof is wrong or that your prof does not agree with it being simpler. Either case would not be an issue though.
Of course! You are in science, not politics. I hope! :-)
It's perfectly ok to tell the professor you found a proof you think it's simpler and much easier to understand. You can do that and still being polite. You can fully express what you have in mind, you don't have to pretend you are asking if it's correct or so, because then you only pass half the message you wanted to pass, the professor might not even be sure why you're sending it and you won't be satisfied with it.
Good professor will respond well. It's of course good to do it in such a way that the communication is just between you two. But possibly better than email is personal consultation, if it's available.
I would think that its perfectly okay to tell the professor that you think you have found an easier way. As long as your polite, and not telling the professor that they are doing a bad job (Or anything like that) then you should be fine. Just make sure you bring it up politely and ask if your easier solution would also work. Professors just want to educate students, so if you find an easier way, let the professor know as soon as possible so that she can teach other students your way, (if it works). That way you will be helping students in the future grasp a topic easier.