This is probably something basic that I am missing. I am reading the article Normal Families: New Perspectives by Lawrence Zalcman, and in one of his examples he makes the following assertion (I am paraphrasing, not quoting, for brevity - I hope I didn't ruin the correctness of the statement):
Let $f_n : D \to \mathbb C$ be a sequence of analytic univalent functions on a domain $D$. Suppose the sequence of first derivatives $g_n := f_n\prime$ converges locally uniformly to an analytic function $g$. Then $g$ is also the first derivative of a univalent function on $D$, or zero.
Why is this true? It looks like a twist on Hurwitz's Theorem, but I don't get it.