I am self-studying real analysis and I am facing a challenge when it comes to create bounds. For example, the 3.12(b) exercise from baby Rudin:
I understood the proof completely. My only problem is that multiplying by $(\sqrt{r_n}+\sqrt{r_{n+1}})$ came from nowhere. I do not know how could I do this by myself.
And this is not special for this example only, I can see it as a general theme when it comes to crafting and engendering these bounds.
Where can I study about these techniques? or is it more of an art than a knowledge?
Any insight?