To each point of $\mathbb Z^2$ is assigned a positive integer

Question

Each lattice point of the plane is labeled by a positive integer. Each of these numbers is the arithmetic mean of its four neighbours (above, below, left, right). Show that all the labels are equal.

I got stuck to this and I really have no idea where to start. Please help.

Answer 1

In any set (finite or infinite) of positive integers there is a smallest value.

Therefore we consider a smallest label $m$. Let $L$ be a lattice point labeled by $m$. Its neighbours are labeled by $a$, $b$, $c$, $d$. Then $m=(a+b+c+d)/4$, or $a+b+c+d=4m$.–––(1).

Now, $a≥m, b≥m, c≥m, d≥m$. If any of these inequalities would be strict, we would have $a+b+c+d>4m$ which contradicts (1).Thus, $a=b=c=d=m$. It follows from this that all labels are equal.

Answer 2

To add to user629660'a (very nice) answer, IF we allowed negative numbers though then This isn't guaranteed to be true. Assign $(a,b)$ the value $f(a,b)= a+b$. So that the numbers are bounded from below OR from above is necessary.

[I didn't see the requirement that $f(a,b)$ be positive until after I wrote my answer. Maybe this still is of interest.]