I am trying to prove the existence of the square root of 2.
The proof: Let
$$S=\{x \in \mathbb{R} ∣x \ge 0, x^2 < 2\}.$$ I understand the proof of LUB, $\alpha$ and so I am at the step where $\alpha^2=2.$
I know that we are to prove by contradiction so we state let $\alpha^2 <2$ and $\alpha^2 >2$. Now my instructor wants us to use the Archimedean Axiom $1/n < \epsilon$.
$(\alpha^2 + 1/n)^2$ then what.....
