Past exam of the integration course of the University of Oxford / 2021 / 3 / a / iii / Solution

Each interval retained at stage $n+1$ lies inside the stage-$n$ interval from which it was cut. Taking the union over all parent components proves $\boxed{C_{n+1}\subseteq C_n}$. Consequently the intersection defining $C$ is a nested intersection of nonempty compact sets.