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**Deadstar** Could someone explain the the conclusion of this proof that the Julia set is bounded to me?

Theorem: The Julia set $\displaystyle J$ of a function $\displaystyle f_c(z) = z^2 + c$ is compact for all $\displaystyle c \in \mathbb{C}$

Proof: First, we show that $\displaystyle J$ is bounded.

Choose $\displaystyle r = \textrm{max}(|c|,3)$ and let $\displaystyle |z| \geq r$.

Then we have,

$\displaystyle |z^2| = |f_c(z) - c| \leq |f_c(z)| + |c|$

And so we have,

$\displaystyle |f_c(z)| \geq |z^2| - |c| \geq 3|z| - |z| = 2|z|$

Hence, since $\displaystyle |z| \geq r$ then

$\displaystyle f_c^n(z) \geq 2^n |z| \rightarrow \infty \textrm{ as } n \rightarrow \infty$ (*)

And so $\displaystyle J(f_c) \subset B(0,r)$. (**)

I found this in a journal, I don't really understand how the last line (**) happens. Surely (*) shows the function diverges... How can we say it's bounded?