The condition of being d bounded means that every point in the space X is within distance r of a. It follows (from the triangle inequality) that any two points in the space are within distance 2r of each other. In particular, every point is within distance 2r of b. So B(b,2r) contains the whole space.

Take any infinite set with thediscrete metric(where d(x,y)=1 whenever x≠y). Then the balls B(x,r) are all disjoint provided that r<1/2. But the space is d bounded because any ball of radius greater than 1 contains the whole space.