Let $\{f_{n}\}_{n}$ be a sequence of absolutely continuous function defined on $[0,1]$ such that $f_{n}(0)=0$ for all $n$. Assume that the sequence of derivatives $\{f_{n}^{`}\}_{n}$ is Cauchy in $L_{1}[0,1]$. Prove that the $\{f_{n}\}_{n}$ converges uniformly to a function $f$, and that $f$ is absolutely continuous in $[0,1]$.
Seriously, I have no idea where to start. This is one of the past qual question. Any help will be much appreciated.