In Rudin's classical "Principles of Mathematical Analysis," he gave a proof like this:
Claim: Let $A= \{p\in \mathbb{Q} | p>0, p^2 <2\}$. Then A contains no largest number.
Proof: Given any $p\in A$. Let $q = p-\frac{p^2 -2}{p+2}$.
Later Rudin claimed that $q^2<2,$ and $q>p$. My instructor asks us to think about a question on our own: Why is such $q$ a natural choice in this proof?
I can see that in this way, $q>p$ is for sure. However, how does it become a natural choice?