# It’s a matter of principles (of convergence)

Let $(f_n)_{n\in \mathbb{N}}$ be a sequence of entire functions such that, for every $z \in \mathbb{C}$, there exists $F(z)=sup_{n\in \mathbb{N}}|f_n(z)| \in \mathbb{R}$. Suppose that $F:\mathbb{C} \to \mathbb{R}$ is continuous and $F(0)=0$. Let $g$ be an entire function such that $g'(0) \neq 0$. Prove that $g \circ f_n \to g$ over compact sets if and only if $f_n(z) \to z$, for every $z \in \mathbb{C}$.