Statement: Let $u_n,u\in W^{k,p}$ where $p\in [1,\infty)$. Then $u_n \rightarrow u$ weakly in $W^{k,p} \Leftrightarrow \partial^s u_n \rightarrow \partial^s u$ weakly in $L^p$ for all $|s|\le k$.
Question. I feel that this statement is true, or at least some variant is. I can prove the $\Rightarrow$ part for $p\in (1,\infty)$, but I'm not so sure about $p=1$ case and the $\Leftarrow$ part. Any help would be greatly appreciated.