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Suppose that $\langle f_n \rangle$ is a sequence of polynomials, $\langlef_n^{(k)}(0)\rangle$ converges for every non-negative integer $k$, and that thelimit 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 areuniformly bounded, then the sequence converges uniformly on compact sets in thecomplex plane. The results imply a theorem of Benz and a conjecture of P\'olya.

The convergence of a sequence of polynomials and the distribution of their zeros