Let $u$ and $v$ are in $W^{1,p}\mathbb{(R^N)}$ then $u,v \in L^{\infty}\mathbb{(R^N)} $ and $uv \in L^P\mathbb{(R^N)}$ for $P>N$
since by Morrey's inequality $||u||_{L^{\infty}}\le ||u||_{w^{1,p}}$ so we can say that $u,v \in L^{\infty}\mathbb{(R^N)} $
but how to prove $uv \in L^P\mathbb{(R^N)}$
I think that you are confused. Clearly if $u,v\in L^{\infty}\mathbb{(R^N)}\cap L^P\mathbb{(R^N)}$ then \begin{equation} ||uv||_{L^p}\le ||u||_{L^\infty}||v||_{L^p}<\infty \end{equation} Indeed: $||uv||_{L^p}^p=\int_{\mathbb{R}^n}u^pv^p \le \int_{\mathbb{R}^n}v^p\sup{|u|}^p = ||u||_{L^\infty}^p\int_{\mathbb{R}^n}v^p=||u||_{L^\infty}^p||v||_{L^p}^p<\infty$ since $u\in L^\infty$ and $v\in L^p$
But maybe you were referring to this result:
$u,v\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ then $uv\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$
You can see section 10.3 of the Giovanni Leoni's book: First course in Sobolev Space (first edition)