If $u_n \to u$ in $L^2(\Omega)$, and $u_n \in L^\infty(\Omega)$, is $u \in L^\infty(\Omega)$?

Question

If $u_n \to u$ in $L^2(\Omega)$, and each $u_n \in L^\infty(\Omega)$, is $u \in L^\infty(\Omega)$ too? There is no uniform bound on $u_n$ in $L^\infty$ though.

I don't think it is.

Answer 1

Hint: Construct a counterexample. Take $u$ to be something unbounded but square integrable (say, $1/x^{1/4}$ in [-1,1]), and take $u_n$ to be truncations at the singularity.