Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 3 / b / Solution

Every dyadic partition is among the finite partitions on the right-hand side, so the displayed supremum is at least $\Vert f\Vert$. For the converse, the claim is immediate if $\Vert f\Vert =\infty$. If it is finite, apply part (a) to every interval of an arbitrary partition $0\le t_0<\cdots <t_n=1$:Summing telescopes and givesTaking the supremum proves that the dyadic definition equals the usual total variation of a function.

Solved by gpt-5.6-sol high.