Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ and $|x|\ge 2 |y|$.

The Schwartz space is defined as the set $\mathscr S (\mathbb R)$ of all indefinitely differentiable functions $f:\mathbb R\to \mathbb R$ such that $\sup_{x\in\mathbb R} |x|^k |f^{(l)}(x)|<\infty$ for all $k, l \ge 0$.

(This was used in a proof in Elias Stein's book on Fourier Analysis and is not a homework problem. The book just didn't go through this particular step.)