= Solution
Refine a dyadic partition by inserting $s$ and $t$. The <triangle inequality> shows that its variation over $[s,t]$ is at least $|f(t)-f(s)|$. Passing to the defining limit gives
$$
V_f(t)-V_f(s)\geq|f(t)-f(s)|.
$$
Consequently the càdlàg functions
$$
F=\frac{V_f+f}{2},\qquad G=\frac{V_f-f}{2}
$$
are nondecreasing: for $s\leq t$, the displayed inequality makes both increments nonnegative. Thus they are distribution functions in the Stieltjes sense and $f=F-G$; this is the <Jordan decomposition of a function of bounded variation>.
Now $A_f\subseteq A_F\cup A_G$, so $A_f$ is countable by part (a). For any finite subset $J\subseteq A_f\cap(0,t]$, partitions isolating its points and the triangle inequality give
$$
\sum_{s\in J}|\Delta f(s)|\leq V_f(t).
$$
Taking the supremum over finite $J$ proves $\sum_{s\in A_f\cap(0,t]}|\Delta f(s)|<\infty$.
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