I'm trying to teach myself some things about Sobolev spaces out of McLean, Strongly Elliptic Systems and Boundary Integral Equations. Exercise 3.14 has me stumped for no reason:
Let $K_j \subset K_{j+1}$ be a sequence of compact sets so that their union is $\mathbb R^n$. Let $\chi_j \in C_c^\infty(\mathbb R^n)$ so that $\chi_j = 1$ on $K_j$. For $s \in \mathbb R$ prove that if $u \in H^s(\mathbb R^n)$ then $\chi_j u \to u$ in $H^s(\mathbb R^n)$.
So far I know that, for $s \geq 0$, $\chi_j u \to u$ in $L^2(\mathbb R^n)$, thus $\hat \chi_j \ast \hat u \to \hat u$ in $L^2(\mathbb R^n)$ but I'm having trouble showing that $(1+|\xi|^2)^{s/2} \hat \chi_j \ast \hat u \to (1+|\xi|^2)^{s/2}\hat u$ which would give convergence in $H^s(\mathbb R^n)$.
Is this even the right approach? What about $s < 0$? I feel like this shouldn't be hard to prove at all but I've been stuck all day.