Assume $A\subset B$ and use the corrected strict identity from part (i). If a walk starts at $y\in A$ and never returns to $A$, then transience makes $L_B<\infty=\widetilde H_A$. Thus
$$
\begin{aligned}
\operatorname{cap}(A)
&=\sum_{y\in A}\mu_yP_y(\widetilde H_A=\infty)\\
&\leq\sum_{y\in A}\mu_yP_y(L_B<\widetilde H_A,L_B\geq0)\\
&=P_{e_B}(H_A<\infty)\\
&\leq\sum_{z\in B}e_B(z)
=\operatorname{cap}(B).
\end{aligned}
$$
This proves monotonicity of the <capacity of a finite set for a transient random walk>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 226 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-226/1/b/ii/solution