Show that if $f$ is Lebesgue integrable and $g$ is Lipschitz on $\mathbb R,$ then $g\circ f$ is Lebesgue integrable.

Question

I need some help with following problem:

Let $f:[a,b]\rightarrow\mathbb{R}$ be a Lebesgue finite integrable function and $g:\mathbb{R}\rightarrow\mathbb{R}$ a Lipschitz function. Show that $g\circ f:[a,b]\rightarrow\mathbb{R}$ is Lebesgue finite integrable function.

Answer 1

Hint: If $g$ is Lipschitz on $\mathbb R,$ then there exist $A,B>0$ such that $|g(x)| \le A + B|x|$ for all $x\in \mathbb R.$