Let $f_n: [0,1] \rightarrow \mathbb{R}$ be a sequence of absolutely continuous functions satisfy the following properties:
$f_n \rightarrow f$ $a.e$ where $f(x) = x$.
$f_n' \rightarrow 0$ $a.e$ and $f_n' \geq 0$ $a.e$ for all $n$.
Is there such a sequence of functions? I could not find such a sequence, so I attempted to disprove it. I realized that using absolute continuity and fundamental theorem of calculus, the first condition gives us:
$\int_{a}^{b} f_n' dm \rightarrow b - a$ for any interval $[a,b] \subset [0,1]$. But I cannot proceed to find the contradiction.