Let $f, g$ be Schwartz functions defined on real numbers. I want to show that for $l\geq 0$, we have $$\sup _x\left|x\right|^l\left|g\left(x-y\right)\right|\le A_l\left(1+\left|y\right|\right)^l$$
In order to show this, we need to observe that $|x|^l|g(x-y)|=|x-y+y|^l|g(x-y)|= \sum^{l}_{k=0}\binom l k |x-y|^k y^{l-k}|g(x-y)|$. Further, we have to note that $|x-y|^k |g(x-y)|$ is bounded as $x$ ranges $(-\infty,\infty)$ because $g$ is Schwartz and there are finitely many functions that are the multiple of $g$ and polynomials.
But the book I am reading (Stein&shakarchi's Fourier Analysis) asserts that I need to consider two cases $|x|\leq 2|y|$ and $|x|\geq 2|y|$. Can someone explain why is that so? I'd appreciate any insight!