The way you have written it, with y constant, it is not true. (Hence chiro's question.) Rather you need something like
$\displaystyle \sum_{k=0}^n a_kx^k \sum_{k= 0}^n b_kx^k= \sum_{k= 0}^{2n} y_kx^k$
That is, you need that "y" different for each power of x and, obviously, if you are multiplying two nth degree polynomials together, you do NOT get an nthe degree polynomial, you get a 2n degree polynomial:
(a+ bx)(c+ dx)= ac+ (ad+ bc)x+ (bd)x^2.
Yes, y is not a constant, that was my mistake.
And...I know i don't get a nth degree polynomial... I edited some time ago to change the picture and replace the n with infinite (I replaced it yesterday)...
And the only thing I have to prove is that the result is a sum of the same form.
This, correcting the $\displaystyle y$ with $\displaystyle y_k$
Again, that certainly is NOT true! Unless you mean, as both chiro and I said before, you mean $\displaystyle \sum_{k=0}^\infty y_kx^k$.
And, in that case, it is pretty close to trivial. The product of any two terms, $\displaystyle a_ix^i$ times $\displaystyle b_jx^j$ must be of the form $\displaystyle a_ib_jx^{j+i}$ so the only thing you can have are "numbers times non-negative powers of x" and that is all the term "$\displaystyle \sum_{k=0}^\infty y_kx^k$ means!