




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 nonpolynomial 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.}