ALIBABA 2021 Problem 7

A subset $Q \subset H^s(\mathbb{R})$ is said to be equicontinuous if for any $\varepsilon>0$ , $\exists \delta>0$ such that $$\|f(x+h)-f(x)\|_{H^s}<\varepsilon, \forall \vert h\vert<\delta, f \in Q.$$ Fix $r<s$ , given a bounded sequence of functions $f_n \in H^s(\mathbb{R}$ . If $f_n$ converges in $H^r(\mathbb{R})$ and equicontinuous in $H^s(\mathbb{R})$ , show that it also converges in $H^s(\mathbb{R})$ .