If $u_n \in C_c^\infty (\mathbb R)$ with $u_n=u(x+n)$ , $n\in \mathbb N$ , $u$ is not identically zero.
How do i prove that $||u_{n+k}-u_n||_{L^q}^q =2||u||_{L^q}^q$.
What my doubt is that even if we take $u_{n+k}$ and $u_n$ to have disjoint support, it doesn't seem that the equality holds . $||u_{n+k}-u_n||_{L^q}^q=\int |u(x+n+k)-u(x+n)|^q$ , i don't see how i can relate the value of $u(x+n+k) $ and $u(x+n)$ looking forward for some hints and help. Thanks