The only technical statements used in this proof are:

1. If a rational function, A(x) / Z(x), vanishes at a, A(x) has (x - a) as a factor. This is easily proven: ide A(x) by (x - a);  if there is a remainder, A(a) is not zero, so that A(a) / Z(a) cannot be zero. If there is no remainder, A(a) has x - a as a factor (note: in our case we have A(x) = R(x) - (c / d)Z(x).)

2. The only rational function that vanishes at infinity and has no singularities in the complex plane is zero.

Proof: The only rational function with no singularities must be a polynomial, since a non-polynomial rational must be singular at each root of its denominator. The only polynomial that  vanishes at infinity is zero because any polynomial behaves at infinity like its highest degree term which is a monomial; _{among monomials only zero vanishes at infinity.}