I need some help with a question from my textbook for dynamical systems with complex numbers.
Let $q_c(z)=z^2+c$.
a) Prove that if $|z|\geq |c|$ and $|z|>2$, then $|q_c(z)|>|z|$ and $z$ is in the stable set of infinity.
b) Use part (a) to show that if $|c|>2$, then $q_c^n(0)$ tends to $\infty$.
The book has an example that if I let $|z|>|c|+1$ I can show that $q(z)=z^{2}+c \geq |z^{2}|-|c| \geq (|c|+1)^{2}-|c|=|c|^{2}+|c|+1$ and then by induction it would follow that this $z$ is in the stable set of infinity since the right side of this equation would get larger by every iteration. Now I'm thinking, in a same fashion, that since $|z|\geq 2>|c|$ it would follow that $|z^{2}+c| \geq |4|+|c|$ and then each iteration would get larger and thus approach infinity. Is But it doesn't feel legit, I feel like I'm missing the point. I would like some suggestions/answers on how to do this..
Thanks in advance
As regards (a), we have that $|z|\geq |c|$ and $|z|>2$ imply $$|q_c(z)|\geq |z|^2-|c|= |z|(|z|-1)-|c|+|z|>|c|(2-1)-|c|+|z|=|z|.$$ For (b), if $|c|>2$ then $f(z):=|q_c(z)|-|z|>0$ for $|z|\geq |c|$. Note that as $|z|\to +\infty$ $$f(z)=|q_c(z)|-|z|\geq |z|^2-|c|-|z|\to +\infty,$$ therefore the non-negative continuous function $f$ has a positive minimum value $a>0$ over the closed set $|z|\geq |c|$. Therefore if $|z|\geq |c|$ then $|q_c(z)|\geq |z|+a\geq |c|$ and $$|q_c^n(0)|=|q_c^{n-1}(c)|\geq |q_c^{n-2}(c)|+a\geq |q_c^{n-3}(c)|+2a\geq |c|+(n-1)a\to +\infty$$ which proves (b).