Imagine an observer on Earth surrounded by a number of objects that are distributed evenly throughout space. Assume that each of these objects have the same intrinsic luminosity, or absolute magnitude.

i) Assuming that these objects are distributed much farther than the limits of your observing capability, derive from geometric principles what behaviour you expect for the number N(f > f_lim) of objects, whose apparent flux is brighter than a limiting flux f_lim. In fact, the assumption of a unique intrinsic luminosity is not required. The objects could just as well be distributed over different luminosities according to some description. As long as that description does not depend on distance, you should find the same number count behaviour.

ii) Swapping apparent fluxes for apparent magnitudes, transform the relation for N(f > f_lim) into one of N(m < m_lim); this is a cumulative number-count relation.