Proof for range of mandelbrot set?

Question

Really simple question, I'm aware that if the value of c used for the mandelbrot set ever gets more than 2 units away from the origin, the value will tend to infinity and it doesn't belong in the set. Is there any proof for this? How do we know that a value greater than 2 will always tend to infinity in the mandelbrot?

Answer 1

If $|c| > 2$, then $f_c(c) = c^2 + c$ has absolute value at least $|c|^2 - |c| = |c|(|c| - 1) > |c|$.

Now, suppose that $|f^{n-1}_c(0)| > |f^{n-2}_c(0)|(|c| - 1)$ and $|f^{n-1}_c(0)| > |c|$ for some $n$.

Then \begin{align*}|f^n_c(0)| &= |f^{n-1}_c(0)^2 + c| \\&\geq |f_c^{n-1}(0)|^2 - |c| \\&> |f_c^{n-1}(0)|^2 - |f_c^{n-1}(0)| \\&= |f_c^{n-1}(0)|(|f_c^{n-1}(0)| - 1) \\&> |f_c^{n-1}(0)|(|c| - 1).\end{align*}

Thus, we inductively have $|f_c^n(0)| > (|c| - 1)|f_c^{n-1}(0)|$ for all $n$, and hence $|f_c^n(0)| > (|c| - 1)^{n-1}|f_c^1(0)| = (|c| - 1|)^{n-1}|c|$, which diverges to infinity, hence $|f_c^n(0)|$ similarly diverges to infinity.