For each $ c\in\mathbb C$, let $ f_c(z,0)=z$, and $ f_c(z,n)=f_c(z,n-1)^2+c$ for $ n\geq1$. a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded. b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.
Problem
Source: Iranian National Olympiad (3rd Round) 2008
Tags: limit, complex analysis, complex analysis unsolved