a) First consider sequence of positive real numbers defined: $ a_0 = A > 0$ $ a_{n + 1} = a_n^2 + |c|, n \geq 0$ If $ (a_n)$ is convergent then it is bounded. So let's assume that $ \lim_{n \to \infty} a_n = g$. Then from the reccurence relation we have: $ g = g^2 + |c|$ $ |c| \leq \frac {1}{4}$ so the above equation has real roots. One of them is $ g_1 = \frac {1 + \sqrt {1 - 4|c|}}{2}$ It's easy to check that if $ a_1 \leq \frac {1 + \sqrt {1 - 4|c|}}{2}$ then $ (a_n)$ is bounded. Now let's take $ |z| \leq \frac {1 + \sqrt {1 - 4|c|}}{2}$ and define new sequence of positive reals: $ a_0 = |z|$ $ a_{n + 1} = a_n^2 + |c|, n \geq 0$ We have $ |f_c(z,n)| \leq a_n \leq \frac {1 + \sqrt {1 - 4|c|}}{2}$ for $ n \in \mathbb{N}$. Thus $ U = \{z: |z| \leq \frac {1 + \sqrt {1 - 4|c|}}{2} \}$ is desired neighbourhood.