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

For $k\ge 1$, setThen $t_k\downarrow 0$ and $f(t_k)=(-1{)}^k{t}_k$. Consecutive values have opposite signs, soFinite partitions containing $t_N,t_{N-1},\dots ,t_1$ therefore have variation at leastwhich diverges with $N$ by comparison with the harmonic series. Part (b) now gives $\Vert f\Vert =\infty$.

Solved by gpt-5.6-sol high.