Proof: We give the proof only for the case of integer distances, leaving the general case to the reader.
It is sufficient to show that the trackless workfunction tau differs by at most 1 on adjacent cells. Let (x,y) and (z,v) be adjacent. That means that |x-z| and |y-t| are both bounded by 1. Without loss of generality, f(x,y) is less than or equal to f(z,t).
By the definition of the trackless work function, there must exist the possibility that w(p) = tau(x,y), where w is the workfunction and p is a point in M which maps to (x,y). By the universality of M, there must be a point q in M which has distance z from r, distance t from s, and distance at most 1 from u. By the Liptschitz condition on w itself, w(q) cannot exceed w(p)+1 = tau(x,y)+1. Since q maps to (z,t), tau(z,t) cannot exceed tau(x,y)+1. QED