Any Requests, Theorem Thursdays Edition?

I don’t know just when I’ll have the energy for my next Mathematics A To Z. But I do want to do something. So for June and July I figure to run a Theorem Thursdays bit. Pitch me some theorems and I’ll do my best to explain what they’re about, or why they’re interesting, or how there might be some bit of mathematics-community folklore behind it. That would be the Contraction Mapping Theorem.

While I’m calling it Theorem Thursdays that’s just for the sake of marketing. It doesn’t literally need to have “theorem” in the thing’s name. The only condition I mean to put on it is that I won’t do Cantor’s Diagonal Argument — the proof that there’s more real numbers than there are integers — because it’s already been done so well, so often, by everyone. I don’t have anything to say that could add to its explanation.

Please, put your requests in comments here. I shall try to take the first nine that I see and feel like I can be competent to handle by the end of July. And I hope I’m not doing something soon to be disastrous. I may not know exactly what I’m doing, but then, if anyone ever did know exactly what they were doing they’d never do it.