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 any dimension, uniform convergence of derivatives and convergence in a point of functions implies uniform convergence of functions (See Baby Rudin). In the analytic case, uniform convergence of analytic functions implies analytic limit using the theorem of Morera.
