I have the following Problem:
Assume that $(u_n)$ is a bounded sequence in $W^{1,p}(I)$ with $ 1 < p \leq \infty $. Show that there exist a subsequence $(u_{n_k})$ and some $ u $ in $ W^{1,p}(I) $ such that $\|u_{n_k} - u\|_{L^\infty} \to 0$. Moreover, $ u'_{n_k} \to u' $ weakly in $ L^p(I) $ if $ 1 < p < \infty $, and $ u'_{n_k} \overset{\star}{\to} u' $ in $ \sigma(L^\infty, L^1) $ if $ p = \infty $.
I am stucked in proving the existence of such $(u_{n_k})$. I know that, since $ W^{1,p}(I) $ is reflexive for $1<p<\infty$, there exist a subsequence $(u_{n_k})$ which converges in the weak topology. But I am trying to see the way to pass from this weak convergenge to strong convergence but I couldn't. Is there any lemma or theorem about the weak topology or the weak convergences sequences in $L^p$ that I am no considering? Any help Will be aprecciated.
