How did Rudin come up with this exact expression? To show that for every rational $p>0$ such that $p^2<2$ one can get a rational $q$ with $p^2<q^2<2$ and $p<q$ Rudin writes,

We now examine this situation a little more closely. Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2>2$. We shall show that $A$ contains no largest number and $B$ contains no smallest.
More explicitly, for every $p$ in $A$ we can find a rational $q$ in $A$ such that $p<q$, and for every $p$ in $B$ we can find a rational $q$ in $B$ such that $q<p$. 
To do this, we associate with each rational $p>0$ the number $$q=p-\frac{p^2-2}{p+2}=\frac{2p+2}{p+2}.$$ Then $$q^2-2=\frac{2(p^2-2)}{(p+2)^2}.$$

This appears many times in analysis, so if the answer can be generalized/a general method can be given then that would be the best thing. My question is how does one come up with the quantity
$$p-\frac{p^2-2}{p+2}$$
My intuition is that can come from any of the rational approximation possibles, by for example using Newton's method, though I'm lacking when it comes to the details.

It's always seemed easier to me, as a matter of intuition, to find $n\in \mathbb N$ such that $(p+1/n)^2<2.$ This is the same as making $(2p)/n+1/n^2<2-p^2.$ And because $1/n^2\le 1/n,$ it suffices to find $n$ such that
$$(2p)/n+1/n= (2p+1)/n<2-p^2.$$ Now you're just a manipulation away from applying the Archimedian principle.
