Fixed-degree polynomial limit

If polynomials of degree at most $d$ converge at $d+1$ distinct points, their coefficients converge, by invertibility of the Vandermonde matrix. Their limit is a polynomial of degree at most $d$, and convergence is uniform on every compact subset of $\mathbb{C}$. Under convergence on compact sets, Rouché's theorem further locates a unique nearby zero for each simple root of the limit, for all sufficiently large indices. Finite initial polynomials need not have any root.