In my searchings for proofs I found this page
Prove the existence of the square root of $2$.
In one of the comments there someone had posted this proof link:
http://www.public.iastate.edu/~roettger/201/sqrt2.pdf
How is this proof even valid? I think it has a "broken twist" at the end of it.
In the proof, $\alpha$ is defined as the least upper bound of the set $A$ of real numbers $x\in\mathbb{R}$ such that $x^2<2$.
It is then shown that $\alpha^2<2$ is not true, and that $\alpha^2>2$ is also not true. By the law of trichotomy (I believe this is the "broken twist" at the end to which you refer), we must have one of
$$\alpha^2>2\ \mathrm{or}\ \alpha^2<2\ \mathrm{or}\ \alpha^2=2.$$
As neither of the first two occur, the third must.