Is it appropriate to email a professor with a simpler solution than the one she provided? 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.