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The ultrafilter proof of Tychonoff's theorem.

The proof is simple, show the power of working with filters and incorporats a good deal of what "everyone should know about compactness".

The strategy-stealing argument for why the first player can force a win in hex.

The argument is simple, elegant, clever and there is essentially no effort in learning it.

The proof of Zorn's lemma by way of ordinals.

Too many people believe that Zorns lemma is an inherently incomprehensible black box. It is not.

Heine-Borel by "induction."

The argument is very neat and shows exactly where the completeness of $\mathbb{R}$ matters.

The visual argument for finding the area of a circle, given radius and circumference.

It's simply beautiful.