Locally uniform limits of univalent functions are univalent or constant

A locally uniform limit of univalent functions can lose injectivity only by collapsing to a constant.Suggest a slogan

Let \(\Omega\subseteq\CC\) be a connected open set and let \((f_k)_{k\geq1}\) be univalent functions on \(\Omega\) converging locally uniformly on \(\Omega\) to \(f\). Then \(f\) is either univalent or constant.