The assertion is true. The standard estimate <half-plane capacity is bounded by squared diameter> gives
$$
\operatorname{hcap}(A_n)\leq C\operatorname{diam}(A_n)^2.
$$
Consequently $\operatorname{hcap}(A_n)\to\infty$ forces $\operatorname{diam}(A_n)\to\infty$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 3 / c / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-220/3/c/ii/solution