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 $|f_{n}'(x)|<M$ for all $x \in \left[0,1\right]$. and all $n \in \mathbb{N}$.Show that $f_{n}\rightarrow f$ uniformly.
I tried to use mean value theorem several times and choose a $y$ very close to particular $x$ and I wrote that $|f_{n}(x)-f(x)|<|f_{n}(x)-f_{n}(y)|+|f_{n}(y)-f(y)|+|f(y)-f(x)|$. However the problem is after showing f is continuous I always choose a $y$ for particular $x$ and In this case I can not go to $sup_{x \in \left[0,1\right]}|f_{n}(x)-f(x)|$