I have a sequence of functions $f_k$ in $L_p(R)$, with $1<p<\infty$ and I'd like to show that it weakly converges to $0$. This is the sequence, where $k\in N$ $$f_k = 1_{[k,k+1]}$$
What I've tried: If $f_k$ converges to $0$, then we should have $$\lim_{k\to \infty}\int_R(f_k-0)\phi dx =0$$ where $\phi \in L_{p'}(R)$ (the dual, or here, $L_q(R)$. Putting in the function, one gets $$\lim_{k\to \infty}\int_k^{k+1}1.\phi dx$$ Now I need to show this goes to $0$ for all functions $\phi \in L_q(R)$$, but that isn't necessarily true right?
With $q = p'$ you can use Hölder's inequality to say that $$\int_{\mathbb{R}}1_{[k,k+1]}|\phi(x)|dx\leq \left(\int_{\mathbb{R}}1_{[k,k+1]}|\phi(x)|^qdx\right)^{1/q}$$ The question is therefore if $$\lim_{k\rightarrow \infty}\int_{\mathbb{R}}1_{[k,k+1]}|\phi(x)|^qdx = 0$$ which you can show using the Dominated Convergence Theorem.