  Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langle
f_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that the
limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2,
\dots$ lie in the closed upper half plane $\rm{Im}\ z\geq 0$, or if $f_1, f_2,
\dots$ are real polynomials and the numbers of their non-real zeros are
uniformly bounded, then the sequence converges uniformly on compact sets in the
complex plane. The results imply a theorem of Benz and a conjecture of P\'olya.
