Under locally uniform holomorphic convergence, zeros persist with their total multiplicity near each zero of the limit.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open and let \((f_k)_{k\geq1}\) be a sequence of holomorphic functions on \(\Omega\) converging locally uniformly on \(\Omega\) to \(f\). Suppose \(f\) has a zero of order \(n\) at \(z_0\in\Omega\). Then there is \(\delta>0\) such that for all sufficiently large \(k\), \(f_k\) has exactly \(n\) zeros in the disc \(\{z : \abs{z-z_0}<\delta\}\), counted with multiplicity. Moreover, for every \(\varepsilon\in(0,\delta)\) there is \(K\) such that for all \(k\geq K\) these \(n\) zeros lie in \(\{z : \abs{z-z_0}<\varepsilon\}\).