Problem #PRU-61457

Problem

Liouville’s discrete theorem. Let $f(x,y)$ be a bounded harmonic function (see the definition in problem number 11.28), that is, there exists a positive constant $M$ such that $\mathrm{\forall }(x,y)\in {\mathbb{Z}}^2$ $|f(x,y)|\le M$. Prove that the function $f(x,y)$ is equal to a constant.