a sequence of holomorphic functions with uniformly convergent derivatives

Let $(f_{n})_{n}$ be a sequence of holomorphic functions on a domain D which satisfies the following conditions: there exists some $z_{0}$ in D such that $f_{n}(z_{0})$ converges and the sequence of $(f_{n}')_{n}$ of their derivatives converges uniformly on compact subsets of D.Then the sequence $(f_{n})_{n}$ itself is also locally uniformly convergent. I try to prove the statement above by using the identity theorem for analytic functions but I couldn't. Thank you for help.

• In your case: the real case implies that $(f_n)_{n\in\Bbb N}$ converges uniformly; now, the complex case implies analytic limit. – Martín-Blas Pérez Pinilla Apr 12 '14 at 13:48
• Open subsets of $\Bbb R^n$ are connected if and only if they are path-connected. Suppose that your result fails in some $z_1$. Take a path from $z_0$ to $z_1$ and a connected compact $K\subset D$ s.t. path$\subset\mathring{K}$. – Martín-Blas Pérez Pinilla Apr 13 '14 at 18:08
• Yes, the uniform convergence of derivatives plus the "anchor" at $z_0$ implies the uniform convergence of functions. – Martín-Blas Pérez Pinilla Apr 13 '14 at 21:52