Density of hyperbolicity in the Mandelbrot set

For each complex parameter $c$, let $f_c : \mathbb{C} \to \mathbb{C}$ be the quadratic polynomial defined by $f_c(z) = z^2 + c$. A point $z \in \mathbb{C}$ is said to be part of an attracting cycle of period $m$ of $f_c$ if $m > 0$, $f_c^{[m]}(z) = z$ (where $f_c^{[m]}$ denotes the $m$-fold iterate of $f_c$), the iterate $f_c^{[m]}$ is complex-differentiable at $z$, and the operator norm of its derivative at $z$ satisfies $\| \mathrm{deriv}(f_c^{[m]})(z) \| < 1$. The Mandelbrot set is the set of all parameters $c \in \mathbb{C}$ for which the orbit of $0$ under $f_c$ does not escape to infinity, or equivalently, for which $\|f_c^{[k]}(0)\| \leq 2$ for all $k \geq 0$. Prove that the set of parameters $c$ for which $f_c$ has an attracting cycle is dense in the Mandelbrot set.