bounds on size of Julia sets inside Mandelbrot set

Question

Let $J_c$ be the Julia set for the quadratic polynomial $f_c(z) = z^2 + c$, and the Mandelbrot set is $M = \{ c \in \mathbb{C} : J_c \text{ is connected} \}$. Call the closed disc of radius $2$ centered at the origin $D = \{ c \in \mathbb{C} : |c| \le 2 \}$, now $M \subseteq D$.

Question: does $c \in M$ imply $J_c \subseteq D$ ?

Answer 1

I think I can get this using the triangle inequality and the fact that all possible $c$ in the mandelbrot set are bounded in the disk of radius $2$.

Let $f_c(z)=z^2+c$ and pick a some $r=|z|>2$.

$|z^2|=|f_c(z)-c| \leq |f_c(z)|+|c|$

which means basically that $|f_c(z)| \geq |z^2|-|c| \geq r|z|-|z|>(r-1)|z|$.

Since $(r-1)>1$, iterates are greater than $r^n|z| \to \infty$ so $z \notin J_c$.