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

Suppose that $⟨f_n⟩$ is a sequence of polynomials, $\langle f_n^{(k)}(0)\rangle$convergesforeverynon-negativeinteger$k$, and that the limit is not $0$ for some $k$. It is shown that if all the zeros of $f_1, f_2, \dots$lieintheclosedupperhalfplane$\rm{Im}\ z\geq 0$,orif$f_1, f_2, \dots$ 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.