Convergence in $L^p(U)$ of smooth functions acted on by a $C^1$ function with $|U|<\infty$

Question

Assume $U$ is bdd and $G:\mathbb{R}\rightarrow \mathbb{R}$ is $C^1$ with $G'$ bdd. If we have smooth $f_n \rightarrow f$ in $L^p(U)$, do we necessarily have that $Gf_n \rightarrow Gf$ in $L^p(U)$?

I feel if $U$ was connected I could just apply the MVT, but as this is not necessarily the case I'm at a loss.

Answer 1

HINT. Establish the following inequality: $$\lvert G(y)-G(z)\rvert^p\le C\lvert y-z\rvert^p,$$ for a constant $C>0$ that is independent on $y$ and $z$.

Answer 2

Forget $C^1$ and smoothness. Just let $G$ be any bounded measurable function on $U$ and assume $f_n \to f$ in $L^p(U).$ Then $\|Gf_n-Gf\|_p \le \|G\|_\infty\|f_n - f\|_p \to 0.$