How does this answer follow? I was looking through proofs of bounds of functions that didn't rely on calculus and one stackexchange topic I came across is this: Show that $f(x,y)=\frac{xy^2}{x^2+y^4}$ is bounded 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 $z=x/y^2$.

The point is that the function $g(z) = z/(1+z^2)$ satisfies $|g(z)| \le 1$ for all $z$. Since, as stated in the question, $f(x,y) = g(x/y^2)$, it must also be true that $|f(x,y)|\le 1$ for all $x,y$.

Note that the top answer is not a full proof of the fact, but rather filling in a gap in the question asker's proof.