A short list of mathematical things I find cool, either because the result is nice or there is a really nice proof.
-Euclid's proof of the infinitude of primes.
A VERY old theorem but the proof is strikingly elegant.
-Prime Number Theorem & Dirichlet's Theorem on Arithmetic Progressions
This was my first exposure to analytic number theory, and how the structure of the primes is encoded in the analytic theory of the Riemann zeta function. It seemed like magic at first. In fact most of complex analysis seemed like magic, so I might as well include Cauchy's Integral theorem in this list.
-Cardinality, Countability.
Arguments like Cantor's diagonal argument and the cardinality proof of the existence of transcendentals are pretty cool.
-Fundamental Theorem of Algebra
How trivial it becomes with the machinery of complex analysis/algebraic topology.
-Brouwer's Fixed Point Theorem
Pretty nice result.
-The existence of irrational a,b such that a^b is rational.
The nonconstructive proof of this by considering root(2)^root(2) is pretty devious.
-Fermat's method of infinite descent
One of my favourite examples of proof by contradiction.
That's it for now, but it is a pretty arbitrary selection of nice results.
