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

Theorem: The Julia set of a function is compact for all

Proof: First, we show that is bounded.

Choose and let .

Then we have,

And so we have,

Hence, since then

(*)

And so . (**)

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