I was looking through proofs of bounds of functions that didn't rely on calculus and one stackexchange topic I came across is this:
The goal is to show $|f(x,y)|$ is less than some real number. The quote on quote "answer", which is extremely presumptuous with almost no reasoning whatsoever, claims
"If $z>1$, then $z^2>z$." Okay, so what?
"If $z \leq 1$, then $z \leq 1+z^2$ since $z>0$." How does that follow and so what?
In math, you are expected to explain your reasoning. So, even if it is right, it's useless unless it can be argued why it makes sense that it is right. We are looking for the largest possible $f$, not of $z=x/y^2$.