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Let $({\mathcal{F}}_n)$ be the natural filtration of the random walk on a graph. For a bounded harmonic function on a graph, the transition rule and the Markov property giveThus $h(X_n)$ is a bounded martingale. The Martingale convergence theorem gives a finite almost sure limit $L$.

On the probability-one recurrence event, every vertex $v$ is visited along a sequence $n_k\to \infty$. Along that sequence $h(X_{n_k})=h(v)$, while convergence of the whole sequence forces $h(v)=L$. Therefore every pair of vertices has the same value. This is the bounded harmonic function theorem on a recurrent graph:Recurrence makes each vertex value a subsequential limit of the same convergent martingale. A connected locally finite graph is countable, since it is the union of its finite distance balls, so these almost sure assertions can also be combined by a countable intersection. The one-vertex case is immediate.