Suppose $f_{n} \to 0$ strongly in $L^{p}_{loc}(\mathbb{R})$ for some $p \in [1, \infty]$, where $f_{n}$ is a smooth sequence. Is it true that $f_{n}' \rightharpoonup 0$ weakly in the sense of measures, i.e. that for any $\phi \in C_{0}(\mathbb{R})$ and compact $K \subset \mathbb{R}$ we have
$$ \int_{K} \phi f_{n}'~dx \to 0 ~?$$
Of course, we can say that $f_{n}' \to 0$ in sense of distributions but I am trying to upgrade this convergence to the above sense. Perhaps there is a clever density argument we can use? Formally I think the result should hold but I am not sure how to prove it.
Thank you!
