TITLE: sequence of absolutely continuous functions, uniform convergence
QUESTION [7 upvotes]: Let $\{f_n\}$ be absolutely continuous functions on $[0,1]$, $f_n \geq 0$ $\forall n \in \mathbb{N}$. 
Suppose that $\lim_{n \to \infty} \int^1_0 f_n dx = 0$. 
Moreover, suppose that $\forall$ $\epsilon > 0$ $\exists$ $\delta > 0$ such that $\int_{E} |f_n'(x)|dx$ < $\epsilon$ $\forall n \in \mathbb{N}$,  if $m(E) < \delta$. 
Prove that $f_n$ converges to $0$ uniformly on $[0,1]$.
Can someone help show this?
Thanking you in advance.

REPLY [2 votes]: Following Mike's comment, pass to a subsequence to get almost everywhere convergence to 0.  Thus there is a set $A$ of measure 1 with $f_n(x) \to 0$ for all $x \in A$.  In particular $A$ is dense.  Now the assumption about the $f'$ guarantees that the family $\{f_n\}$ is equicontinuous (since $|f(x)-f(y)| \le \int_x^y |f'(t)|\,dt$).  One can now follow the proof of the Arzela-Ascoli theorem to obtain the uniform convergence.