Let $R: \Omega\times(t_{0}, \infty)\to \mathbb{R}$ be a function with $\Omega$ being a domain of $\mathbb{R}^{d} (d\geq 1)$. Prove that
If $R(t)\to 0$ in $L^{2}(\Omega)$ as $t\to \infty,$ then $R(x, t)\to 0 \;a.e.$ in $\Omega\times(t_{0}, \infty)$ as $t\to \infty.$
How could I do this using sequences?
Thank you in advance.
Surely false. Take any sequence $f_n$ which tends to $0$ in $L^{2}$ but not a.e. Take $R(x,t)=f_n(x)$ if $n<t <n+1$ and $0$ if $t \notin (n,n+1)$.