I believe I have a new method of proof for an already existing theorem. The theorem itself is quite elementary and the standard proof for it, that I have seen in several books, is not difficult. Nonetheless, I believe my proof is simpler yet. My question is: How do I find out if my proof has ever been published before? In addition, I have a proof for what is a new theorem to me in the sense that I have never seen it before. How do I find out if it is really new? Moreover, I don't know if it really rises to the level of new "theorem". It's also quite elementary and the kind of thing that could be found in a textbook in the exercises. How do mathematicians tell the world when they have a new but elementary theorem?