If $u_n \rightharpoonup u$ in $L^2(\Omega)$, does $u_n^+ \rightharpoonup u^+$ in $L^2(\Omega)$?

Question

Let $\Omega$ be a bounded domain.

If $u_n \rightharpoonup u$ in $L^2(\Omega)$, does $u_n^+ \rightharpoonup u^+$ in $L^2(\Omega)$ where $u_n^+ = \max(0,u_n)$.

Note all convergences are weak.

My guess is no though I hope the answer is yes. I'm not sure at all how to prove it.

Answer 1

To summarize the comments: the answer is negative, and functions like $\sin nx$ or Rademacher functions $r_n$ provide counterexamples. They converge weakly to $0$, but the integral of the positive part does not tend to $0$.