Let $(g_n)_n$ be a sequence of holomorphic functions on $U$, where $U$ is the open unit disk. Suppose the first $k$ derivatives of $g_k$ at zero all vanish, $g_k(0) = 0$, and finally that $g_n$ converges pointwise to a holomorphic function $g$ on $U$.
Must $g$ be the zero function? (I'm guessing yes. It would for example suffice to show that $g_k'$ converges pointwise to $g'$.)