Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and $\{u_n\}$ converges to $u$ in the norm on $L^p$, then upto subsequence $\{u_n\}$ converges to $u$ pointwise a.e. in $\Omega$.
Is this statement true? Please reply with an explanation... Thanks...
Say, $\|u_{n_{k}}-u\|_{L^{p}}<\dfrac{1}{k^{2}}$, $k=1,2,...$, $(n_{k})$ strictly increasing, then \begin{align*} \int\sum_{k}|u_{n_{k}}-u|^{p}=\sum_{k}\int|u_{n_{k}}-u|^{p}=\sum_{k}\|u_{n_{k}}-u\|_{L^{p}}^{p}\leq\sum_{k}\dfrac{1}{k^{2p}}<\infty, \end{align*} so for a.e. $x$, \begin{align*} \sum_{k}|u_{n_{k}}(x)-u(x)|^{p}<\infty, \end{align*} so $u_{n_{k}}(x)\rightarrow u(x)$.