Let $ f_n: [0,1] \to \mathbb R $ be a sequence of derivable functions with $|f_n'(x)|<3$ $\forall x\in[0,1]$ and $\forall n\in\mathbb N$.
Assume that the series is pointwise converge to $L(x)$ on $[0,1]$. Prove that the series is uniformly converge to $L(x)$ in $[0,1]$.
Any help would be greatly appreciated.