Let $f \in C^{\infty}_{c}(\mathbb{R})\setminus \lbrace 0 \rbrace $ with $supp(f)\subset [0,1]$ and $p \in (1,\infty)$
Define $f_{n}(x) = f(x-n) , n \in \mathbb{N}$
So I start with $\lim \limits_{n \to \infty}(\int_n^{n+1} (f_{n}g) dx)\leq \lim \limits_{n \to \infty}(\int_n^{n+1} |f_{n}|^p dx)^{1/p}(\int_n^{n+1} |g|^{p^{`}} dx)^{\frac{1}{p^{´}}}$ with $g \in L^{p^{´}}$
Then $(\int_n^{n+1} |f_{n}|^p dx)^{1/p}= C$ .
So I have to show $\lim \limits_{n \to \infty}(\int_n^{n+1} |g|^{p^{`}} dx)^{\frac{1}{p^{´}}} = 0$
But if I´m right then I dont know how to show the last equation. Can someone help me there, please?
$\int_n^{\infty} |g|^{p'} \to 0$ by DCT since $I_{(n, \infty)} \to 0$ and $|g|^{p'} $ is an integrable dominating function. Hence $\int_n^{n+1} |g|^{p'} \to 0$.