= 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.
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