If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?

I'm not sure if this question is trivial or not, and how to even start! Any hints would be great!

I would add that $\Omega$ is a bounded open subset of $\mathbb{R}^n$.

If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?

I'm not sure if this question is trivial or not, and how to even start! Any hints would be great!

Edit 1: I would add that $\Omega$ is a bounded open subset of $\mathbb{R}^n$.

Edit 2: I want to apologize for the confusion caused by the symbol "$\rightharpoonup$", which means weak convergence. I used it under the assumption that it is standard.

Thank you for the counter-example and the answer. Bill Johnson mentioned in the comments that the sequence $\{t\mapsto \sin(nt)\}$ converges weakly in $L^2(0,T;L^2(0,1))$, but no subsequence has the sought after property.

Which urges the question: what is a non-trivial sufficient condition so that the question has a positive answer?