Question: Let $P$ be a polynomial of at least degree $2$. Using Vitali's theorem show that $P^n\to \infty$ on the component $F_\infty$ of $F(P)$ which contains $\infty$.

Relevant theorem:

Theorem (Vitali): Suppose that the family $\{f_1,...,f_n\}$ of analytic maps is normal in a domain $D$ and that $f_n$ converges pointwise to some function $F$ on a non-empty open subset $W$ of $D$. Then $f$ extends to a function $F$ which is analytic on $D$ and $f_n \to F$ locally uniformly on $D.$

Sketch of proof: Let $W$ be a non-empty open subset in $F_\infty$. In my mind it's pretty clear that on every nonempty subset $W$ we know that $P^n(z)$ will pointwise converge to $h(z)$ where $h(z)=\infty$ for all $z$. Hence $P^n(z) \to h(z)$ on all of $F_\infty$, so $P^n(z) \to \infty$ on $F_\infty$.

I'm stuck on how to apply Vitali's theorem in this case. I've tried sketching out what I'm thinking, but I can't convince myself that I'm even on the right track.


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