Showing weak convergence of a sequence in $L^p(R)$ 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.
