Proving a function $f(m,n)$ which satisfies two conditions is a constant I found the following question in a book only with one sentence. "This question can be solved by an elementary way. Note that the following two are false: (1) If a function is bounded from below, then it has minimum value. (2) A monotone decreasing sequence reaches a negative value."
Question: Let $m,n$ be integers. Supposing that a function $f(m,n)$ defined by $m,n$ satisfies the following two conditions, then prove that $f(m,n)$ is a constant.
1. $f(m,n)\ge0$.
2. $4f(m,n)=f(m-1,n)+f(m+1,n)+f(m,n-1)+f(m,n+1)$.
I suspect this question can be solved by a geometric aspect. I've tried to prove this, but I'm facing difficulty. Could you show me how to prove this?

A probabilistic proof is given at the bottom of this page.
