Let $f_{n}:\left[0,1\right]\to\mathbb{R}$ be a sequence of differentiable functions converging to $f:\left[0,1\right]\to\mathbb{R}$ pointwise. Assume that there exists a constant $M>0$ such that $|f_{n}'(x)|\le M$ for all $x\in \left[0,1\right]$ and all $n \in \mathbb{N}$. Show that $f_{n}\to f$ uniformly.
My attempt was I assumed that let $f_{n}\to f$ not uniformly. Then there exists an $\epsilon >0$ such that for all $N_{1}>0$ there exists an $n_{1}\ge N_{1}$ and there exists an $x_{1} \in \left[0,1\right]$ such that $|f_{n_{1}}(x_{1})-f(x_{1})|\ge \epsilon $. Moreover for all $x,y\in \left[0,1\right]$, $|f_{n}(x)-f_{n}(y)|\le M|x-y|$ by using mean value theorem for $x$ and $y$. I can see a sequence here like $f_{n_{k}}(x_{k})$ but I could not proceed after this point. I could not understand the answer well for the same question and I am asking if I can do this with my own way and if anyone can proceed?