Show that if $u,v\in H^s(\mathbb{R}^n)$ for $s>{n\over 2}$, then $uv \in H^s(\mathbb{R}^n)$ and $$ \|uv\|_{H^s(\mathbb{R}^n)} \le C\|u\|_{H^s(\mathbb{R}^n)} \|v\|_{H^s(\mathbb{R}^n)}, $$ the constant $C$ depending only on $s$ and $n$.
Progress
I tried to prove by induction on the order of the derivatives, but to use the inductive hypotheses could not get the term right inequality. My problem is to find the appropriate method to be able to solve actual inequality.
