I am trying to understand the proof above, Proposition 7, given by the author, but I got stuck at some points.
My explanation:
Now, suppose for contradiction there is an $\epsilon > 0$, for all $x > 0$ such that $NOT (x^2 < 2 < (x+ \epsilon)^2)$,
Since $NOT (x^2 < 2 < (x+ \epsilon)^2) \ \ iff \ \ (x^2 \geq 2 \ \ OR \ \ (x+ \epsilon)^2 \leq 2)$,
We must show that for some $\epsilon>0$ and for all $x > 0$, the statement $x^2 \geq 2 \ \ OR \ \ (x+ \epsilon)^2 \leq 2$ leads to a contradiction.
I think in the proof above, author tries to get a contradiction using right hand side of $OR$ clause, namely $(x+ \epsilon)^2 \leq 2$. But no contradicition is achieved by him for the left hand side side of $OR$ clause namely, $x^2 \geq 2$.
First Question is, am I correct at this point, i.e. the need for a contradiction for the left hand side side of $OR$ clause?
Second question is, I don't understand how he derived the following sentence in his proof above,
Since $0^2 < 2$, we thus have $\epsilon^2 < 2$, which then implies $(2 \epsilon)^2 < 2$ and indeed $(n \epsilon)^2 < 2$
Why does he feel a need to multiply $\epsilon$ constantly by $n$