A holomorphic function with zeros accumulating inside a connected domain must vanish identically.Suggest a slogan
Let \(\Omega\subseteq\CC\) be a connected open set and let \(f\) be holomorphic on \(\Omega\). If the set of zeros of \(f\) has a limit point in \(\Omega\), then \(f\equiv0\) on \(\Omega\).