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Im new to analysis and in a lecture my professor told us that we can by induction show that if $|c|>2$ then a sequence of $z_n$'s is not bounded, where for every $c\in \mathbb{C}$ $P_c(z)=z^2+c$ is a function and a sequence of complex numbers is defined recursive as $z_1=0$, $z_2=P(z_1)=c$, $z_{n+1}=P(z_n)$.

The hint to show this is: Assume $|c|=2 + \delta$ with $\delta >0$. Use induction to show that $|z_n|\geq 2+(n-1)\delta$ for $n \geq 2$.

Can someone help me with some more hints? Maybe with what is my basis, induction hypothesis and induction step to show this?

Edward P. Kanverapaij
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    For the induction step, start with $|z_{n+1}| = |z_n^2 + c| \ge |z_n|^2 - |c|$. – Martin R Sep 15 '21 at 08:49
  • Do you understand a proof by induction generally? What is unclear about the basis and the induction hypothesis? – Martin R Sep 15 '21 at 08:52
  • I understand the idea of induction but its the first time Im trying to proof something by induction, so to start with I thought I had to show the basis, but I don't know what the basis is in this proof. – Edward P. Kanverapaij Sep 15 '21 at 08:58
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    Check this: https://math.stackexchange.com/a/961486/42969 – Martin R Sep 15 '21 at 09:03

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