Proofs from the "Ugly Book"

Question

There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems.

If there is a book written down by God, why not a book from the Devil? I mean, a book with the most ugly proofs, but yet the best ones we have as accepted proofs. I don't mean to make a horrible proof on purpose, but sometimes ugly proofs is all you have.

I wish if you could share some theorem from the Ugly Book, some theorem proved by a real ugly proof (and yet the only one that there is). I'm asking this not for fun only, but I'm curious about how ugly proofs can be.

Answer 1

Beauty is quite subjective. One may prove something in 2 lines using the newly developed supersymmetric coffee spaces, and that may be cool to some. Suppose you have another proof of the same result that uses only some primitive set of axioms. This proof may potentially be 1000 pages long, but it will be more beautiful to some (for example me), as it is a demonstration of the fact that all that mind blogging complexity is actually the result of addition, multiplication, etc, and some first order logic.

Answer 2

There exists irrational numbers $x$ and $y$ such that $x^y$ is rational.

Proof: If $\sqrt{2}^\sqrt{2}$ is rational, we can take $x=y=\sqrt{2}$. If $\sqrt{2}^\sqrt{2}$ is irrational, we take $x=\sqrt{2}^\sqrt{2}$ and $y=\sqrt{2}$.

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The proof is based on a case distinction in which only one case is true, without telling us which one. The proof is discused at this wikipedia page.