Hah, I didn't notice you put it in the submission, or I probably wouldn't have written out the proof. :P I like to show this to low-level math classes (I'm a grad student, so I end up teaching freshmen a lot) with two different colors of chalk on the board. Inevitably, at least one person thinks it's pretty cool, which makes it worth it.
BTW, the infinite Ramsey theorem and the finite Ramsey theorem are equivalent. Obviously, the infinite version implies the finite version. To show the finite version implies the infinite version, you can use the countable version of Zorn's lemma (which you prove from the countable theorem of choice -- note it's a theorem, not an axiom).
BTW, the infinite Ramsey theorem and the finite Ramsey theorem are equivalent. Obviously, the infinite version implies the finite version. To show the finite version implies the infinite version, you can use the countable version of Zorn's lemma (which you prove from the countable theorem of choice -- note it's a theorem, not an axiom).