Density of hyperbolicity for Multibrot sets

For an integer $n \geq 2$, the Multibrot set of power $n$, denoted $\mathcal{M}_n$, is the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $f_c(z) = z^n + c$ does not escape to infinity. Equivalently, $\mathcal{M}_n$ is the set of all $c \in \mathbb{C}$ such that $\|f_c^{[k]}(0)\| \leq 2^{(n-1)^{-1}}$ for all $k \geq 0$, where $f_c^{[k]}$ denotes the $k$-fold iterate of $f_c$.

A point $z \in \mathbb{C}$ is said to be part of an attracting cycle of period $m$ for a holomorphic map $f: \mathbb{C} \to \mathbb{C}$ if $m > 0$, $f^{[m]}(z) = z$, the iterate $f^{[m]}$ is differentiable at $z$, and $\|\operatorname{deriv}(f^{[m]})(z)\| < 1$.

Prove that the set of parameters $c \in \mathbb{C}$ for which $f_c(z) = z^n + c$ has an attracting cycle is dense in $\mathcal{M}_n$. That is, prove that

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