$L^p$ Convergence

Question

I need to show that if $\{u_n\}\subset L^p(\Omega)$ converges to some $u\in L^p(\Omega)$ and $G\in C^1(\mathbb R)$ with $G(0)=0$ and $|G'(s)|\le M$ then $G(u_n)$ converges to $G(u)$ in $L^p(\Omega)$.

I tried it but fails to solve it. Any type of help will be appreciated. Thanks in advance.

Answer 1

By the mean value theorem, $|G(x)-G(y)|\leq M|x-y|$. Then, $$\int_\Omega|G(u_k(x))-G(u(x))|^pdx\leq M^p\int_\Omega|u_k(x)-u(x)|^pdx=M^p\|u_k-u\|_{L^p}^p$$

Answer 2

$|G'(s)| \leq M \implies |G(s)-G(y)| \leq M|x-y|$

Thus :

$$\int_\Omega|G(u_n)-G(u)|^p\leq M^p\int_\Omega|u_n-u|^p \to 0.$$

Furthermore $G(u_n)$ and $G(u)$ are in $L^p$. Indeed :

$$\int_\Omega|G(u_n)|^p=\int_\Omega|G(u_n)-G(0)|^p \leq M^p\int_\Omega|u_n-0|^p <\infty.$$

Same proof for $G(u)$.