Solution
= Solution
If $A\subseteq C$, Brownian motion exits $\mathbb H\setminus C$ no later than it exits $\mathbb H\setminus A$. The <Strong Markov property> and the nonnegative harmonic function $h_A$ show that the expected exit height for $C$ is at least that for $A$. Taking the limits in the <Brownian representation of half-plane capacity> proves the <monotonicity of half-plane capacity>
$$
\operatorname{hcap}(A)\leq\operatorname{hcap}(C).
$$