Let $(f_n)$ be a sequence of functions on $[a,b]$ such that each $f_n$ is continuous on $[a,b]$ and differentiable on $(a,b)$ and there exists $M>0$ such that $|f_n'(x)|≤M$ for all $x \in (a,b)$ and for all $n \in \mathbb{N}$.
If $(f_n)$ converges pointwise on $[a,b]$, show that $(f_n)$ converges uniformly on $[a,b]$.
I have tried to do integration on both sides of $|f_n'(x)|≤M$, and conclude $f_n$ is Lipschitz continuous.