1) Proceed by induction on the degree of . The statement is obviously true for degree 0. Now assume it is true for degree . Let be a polynomial of degree . Then it can be written as
for some polynomial with degree (or less). Similarly, let
where is the conjugate polynomial of . Now,
By induction, has real coefficients. Next, by the multiplicative property of conjugation, is still the conjugate of . Thus, when added, the imaginary parts cancel out. Lastly, is obviously real. Thus, must also have real coefficients.
2) If for some integer and odd integer , then the infinite product certainly does not converge as for all . Now if for some , then is 0, so the product diverges. Finally, if for , then the product converges to 1 if is even and -1 if is odd.