Let $f$ be a polynomial of degree greater than 1. Prove that the following set is compact:
$K(f)= ( z \in \mathbb{C}: f(f(f(...f(z)...)))$ does not diverge )
Alternatively, the set of all complex numbers such that an infinite number of iterations by $f$ does not diverge ( not necessarily converge )
My attempt:
Setting this proof up I went with the “closed and bounded” approach. I believe I managed to get boundedness by appealing to the fact that after each iteration the degree of the polynomial will grow fast. However proving this set is closed has alluded me; I tried setting up a sequence that converges to a point and showing that point is in $K(f)$. While doing this I tried using the fact that each iteration is a polynomial, and hence, continuous, but this doesn’t give a straightforward explanation as to why the infinite iteration will not diverge.
Background: Complex Qual question #$7$ (www.math.tamu.edu/graduate/phd/quals/ncomplex/a17.pdf)