Under what conditions does a sequence $(f_n)$ in $H^s_0(G)$, that converges in $L^2(G)$, where $G\subseteq \mathbb{R}$ is bounded with smooth boundary imply convergence in $H_0^s(G)$ where $s>2$?
You can furthermore assume that $\sup\limits_{n\in \mathbb{N}} \|f_n\|_{H^s}<\infty$.
Where $\|\cdot\|_{H^s}=\|(1+|\xi|^2)^{s/2}\mathcal{F}(\cdot)(\xi)\|_{L^2(\mathbb{R},d\xi)}$ and $\mathcal{F}$ is the Fourrier transform on $L^2(\mathbb{R})$.
$H^s_0(G)$ is the closure of all testfunctions with compact support in $G$ wrt. the above norm.