Dyadic approximation of total variation (${\sum }_{k<\lceil 2^{n}t\rceil }|X_{(k+1)2^{-n}}-X_{k{2}^{-n}}|\to {\mathrm{Var}}_{[0,t]}X$)

For a càdlàg function, replace the last sampled value at $2^{-n}\lceil 2^{n}t\rceil$ by $X_t$. Right continuity makes this change tend to zero. The corrected sum is bounded by the total variation of a function. Conversely every finite partition of an interval can be approximated by dyadic points from the right, retaining the endpoint $t$. The triangle inequality bounds that partition's limiting sum by the dyadic sums. Taking the supremum proves convergence, including the value $+\infty$.