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IlyaShpitser comments on Open thread, Oct. 5 - Oct. 11, 2015 - Less Wrong Discussion
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FYI, this is called the sum of degrees theorem. In fact, the sum of degrees is not only an even number, but twice the number of edges in the graph. This is due to Euler, I think. He used the famous Koenigsberg bridges problem as a motivation for thinking about graphs.
Good work on thinking about proofs, +1 to you.
I love that I can come to this website and have one of Judea Pearl's former students check my elementary graph-theoretic proofs.
But really, thanks for the encouragement. I had also been wondering if it had a name.