Problem Statement: prove that every pair of real polynomials satisfies some real polynomial relation .
Let, without loss of generality,
Then can be written as a linear combination of . So, in , we write as when , and replace by . By repeating this process as long as there are powers of greater than or equal to in , we shall eventually find an equation where the highest power of is no greater than . Now let the highest power of in this equation be . Then in a similar fashion, we can plug in the value of in to obtain an equation where the power of will not exceed . By repeating this process, we shall eventually obtain an equation where the power of is zero, and that is our desired .