I'm reading the book Grigis, Sjöstrand, Microlocal Analysis for differential operators. Introducing fractional Sobolev spaces, it says that for all test functions $\phi\in C^\infty_0$ and any $u\in H^s$ (for $s\in \mathbb{R}$) we have the inequality $\Vert \phi u\Vert_{H^s}\leq C_{\phi}\Vert u\Vert_{H^s}$ for a constant $C_{\phi}$.
I know the proof for $s\in \mathbb{N}$ but not for general $s\in\mathbb{R}$. I would appreciate any help.