*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.