Discrete harmonic function

A function on the vertices of a locally finite graph is discrete harmonic when its value at every vertex is the average of its values at the neighbouring vertices.

Every bounded discrete harmonic function on ${\mathbb{Z}}^d$ is constant. After rescaling its range to $[0,1]$, place it in the compact convex set of all such functions; translation invariance and the mean-value identity force every extreme point to be constant, and the Krein-Milman theorem finishes the proof.