# Norm of compactly supported functions with disjoint support .

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

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If the supports of $v$ and $w$ are disjoint it holds that $$\int |v + w|^q = \int_{\mathrm{supp}(v)}|v+w|^q + \int_{\mathrm{supp}(w)}|v+w|^q.$$ Since $w$ is zero on $\mathrm{supp}(v)$ it holds that $$\int_{\mathrm{supp}(v)}|v+w|^q = \int_{\mathrm{supp}(v)}|v|^q$$ and vice versa.