= Dyadic approximation of total variation
{title2=$\sum_{k<\lceil2^nt\rceil}|X_{(k+1)2^{-n}}-X_{k2^{-n}}|\to\operatorname{Var}_{[0,t]}X$}
For a <càdlàg function>, replace the last sampled value at $2^{-n}\lceil2^nt\rceil$ 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 $+\infty$.
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