If $a_n \to a$ in $L^2$ and $F:\mathbb{R} \to \mathbb{R}$ is continuous, does $\int F(a_{n_j}) \to \int F(a)$?

Question

Suppose $a_n \to a$ in $L^2(\Omega)$. Let $F:\mathbb{R} \to \mathbb{R}$ be continuous with $F(0) = 0$. We have that $F(b) \in L^1(\Omega)$ if $b \in L^2(\Omega)$ and $|F'(x)| \leq C_1 + C_2|x|$.

I want to show$$\int_{\Omega}F(a_{n_j}) \to \int_{\Omega}F(a)$$ for a subsequence.

Is it possible?

Answer 1

Since my attempt to close the question as a duplicate failed, I'll post an answer: the statement follows from a general theorem on the continuity of Nemytskii operator, which is stated and proved here.