绝对连续且有限变差函数的变差函数绝对连续性证明问询

我现在想证明下面这个命题：

Proposition
If $F:\mathbb{R}\to\mathbb{R}$ is absolutely continuous and of finite variation, then $V_F$ is absolutely continuous.

这是我目前的尝试：

Let $\epsilon>0$. By the absolute continuity of $F$, we can choose a $\delta$ such that $\sum_{i=1}^n|F(t_i)-F(s_i)|<\epsilon$ holds whenever ${(s_i,t_i)}{i=1}^n$ is a finite sequence of disjoint open intervals for which $\sum{i=1}^n(t_i-s_i)<\delta$. Then each finite sequence ${(u_j,v_j)}_{j=i}^m$ of disjoint open subintervals of ${(s_i,t_i)}$...

备注：内容来源于stack exchange，提问作者Beerus