= Solution
The <Strong Markov property> at the first visit to $0$ gives the <singleton equilibrium potential from the Green function>
$$
h_x=P_x(\tau<\infty)=\frac{g(x,0)}{g(0,0)}.
$$
Let $U_n$ be an increasing exhaustion of $G$ by finite sets containing $0$, and let $g_{U_n}(x,0)$ be the <Green function of a transient weighted graph> stopped on leaving $U_n$. Each $g_{U_n}(\mathord\cdot,0)$ has finite support. The Green identity and the <Markov property> give, for $m\geq n$,
$$
\mathcal E\bigl(g_{U_m}(\mathord\cdot,0)-g_{U_n}(\mathord\cdot,0),
g_{U_m}(\mathord\cdot,0)-g_{U_n}(\mathord\cdot,0)\bigr)
=g_{U_m}(0,0)-g_{U_n}(0,0).
$$
By the <monotone convergence theorem>, the right side tends to zero as $m,n\to\infty$, while $g_{U_n}(x,0)\uparrow g(x,0)$. Thus $g(\mathord\cdot,0)$ is an $H_0$ limit of finitely supported functions. Dividing by the positive number $g(0,0)$ proves $h\in H_0$.
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