How to show that every bounded variation function on $[a,b]$ is differentiable a.e? I'm struggling with this question for over a week now. I know the proposition is true, but haven't managed to prove it yet. any suggestions anyone?
($f$ is BV on $I$ if $$\sup\left\{\sum|f(b_k)-f(a_k)| :a_{k+1}\gt b_{k}\gt a_{k} ; a_k,b_k\in I\right\}\lt \infty)$$ 

Every function of bounded variation is the difference of two monotonically non-decreasing functions. Monotonically non-decreasing functions are differentiable almost everywhere.
