Mandelbrot: if $|c|>2$ the set is not bounded

Question

First of all: I'm very new to both the Mandelbrot Set, math and analysis in general - please be gentle. :-)

I was given the Mandelbrot set as follows: For any $c\in \mathbb{C}$ the function $P_c(z)=z^2+c$ is defined.

1. Show by induction that if $|c|>2$, then the set is not bounded. Hint: Assume that $|c|=2+\delta$ where $\delta >0$. Use induction to show that $\left|z_{n}\right| \geq 2+(n-1) \delta$ for $n\geq 2$.
2. Conclude that the set $M=\{c\in \mathbb{C}:$ the sequence of $z_n$ is bounded$\}$ is a subset of $\{z \in \mathbb{C}:|z| \leq 2\}$.

I've only just learned the induction method, but is unable to see how i would use it to solve 1., but any hints or points in the right direction would be appreciated! :)

Note: I've found this stackexchange post Show that Mandelbrot set is contained within the closed disc of r=2, where a similar proof is conducted only using $|z_n|\geq 2+(2^n-1)\varepsilon$ instead, but I'm unsure if i can use some of the same math here.

Answer 1

There are some similarities in the post you linked to your problem here that you can try to adapt into a proof. I'll give a sketch about what you can do for (1), assuming that you are new at analysis, and let you think about the rest.

We want to look at estimating $|z_n|$ where $z_{n + 1} = z_n^2 + c$ and $z_1 = 0$. Following the hint, let $\delta > 0$ and assume that $|c| = 2 + \delta$. Then the base case $n = 2$ holds easily, so we are left with the inductive step. Suppose that $|z_k| \geq 2 + (k - 1)\delta$ holds for some $k \geq 2$. Then $$|z_{k + 1}| = |z_k^2 + c| \geq ||z_k|^2 - |c||$$ (where did the inequality come from?) Now, by the inductive hypothesis, we have $$|z_k|^2 \geq (2 + (k - 1)\delta)^2 = 4 + 4(k - 1)\delta + (k - 1)^2\delta^2 \geq 4 + 4k\delta - 4\delta \geq 4 + k\delta + \delta$$ where the last inequality holds for all $k \geq 2$ (once again, why do the last two inequalities hold?) Combining both these lines gives $$|z_{k + 1}| \geq 4 + k\delta + \delta - 2 - \delta = 2 + ((k + 1) - 1)\delta,$$ so by induction, the inequality holds for all $n \geq 2$. This shows that $|z_n| \to \infty$ as $n \to \infty$ whenever $|c| > 2$ and so $|z_n|$ cannot possibly be bounded for all $n \geq 1$.