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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 givesConsequently the càdlàg functionsare nondecreasing: for $s\le t$, 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 giveTaking the supremum over finite $J$ proves ${\sum }_{s\in A_f\cap (0,t]}|\Delta f(s)|<\infty$.

Apply part (a)'s square formula, extended by the Jordan decomposition of a function of bounded variation from nondecreasing functions to arbitrary càdlàg functions of bounded variation, to $f+g$, $f$, and $g$. Sincelinearity of the Lebesgue-Stieltjes integral leaves ${\int }_0^{t}fdg+{\int }_0^{t}gdf$. At each time $s$, polarization of the jump correction givesOnly common jump times contribute, and hence