If i have a sequence $u_n=u(x+n)$ , $u_n\in C_c^\infty R$, is it bounded in $W^{1,p} (R)$ ? and is it true that for no $q\ge1$ does there exist a subsequence converging strongly in $L^q(R)$ .
As per my lecture notes the claim is that $\int_R u_n\phi\to 0$ and $\int_R u_n'\phi\to 0$ as $n\to \infty$ . I am not able to clarify these statements . Any explanations, hints etc would really help me .