Let $u,v\in W^{1,p}(\mathbb{R}^n)$, and $p > n$. Show that $uv \in W^{1,p}(\mathbb{R}^n)$, and moreover that $uv$ satisfies the inequality: $$ \|uv\|_{W^{1,p}} \leq C\|u\|_{W^{1,p}}\|v\|_{W^{1,p}} $$
The idea here seems to be to use the Sobolev inequality (Gagliardo Nirenberg) to get some sort of quantitative estimate on $\|uv\|$, however I am even having issues showing the product is in $L^p$. A small hint or idea would be appreciated. I also think we may want to approximate $u,v$ with test functions, as we have the full space $\mathbb{R}^n$ to our disposal.
Notice that the terms you have to bound look like $L^p$ norms of $uv$, $v\partial_i u$ or $u\partial_i v$ for some $i=1,\ldots,n$. To do this just recall that for $p>n$ and $w\in W^{1,p}(\mathbb R^n)$ we have $w\in L^\infty(\mathbb R^n)$ and $$ \| w\|_{L^\infty}\lesssim \| w\|_{W^{1,p}}. $$