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

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 $L_B<\infty ={\tilde{H}}_A$. ThusThis proves monotonicity of the capacity of a finite set for a transient random walk.