Bounded harmonic function theorem on the integer lattice
= Bounded harmonic function theorem on the integer lattice
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.