I know there is a very well known proof that for any rational such that
$$x=\frac p q < \sqrt 2$$
there exists another rational $y=\dfrac mn$ such that
$$x=\frac p q < \frac m n <\sqrt 2$$
Happily I've forgotten most of the proof, which lets me try and build it myself.
Could someone provide a hint to produce a proof? I guess I should start with $2p<3q$ to produce a number larger than $p/q$ but smaller than $\sqrt 2$.
In parenthesis, would this prove that $\sqrt 2 \notin \mathbb Q$?