Local connectivity of Multibrot sets

For every natural number $n$, the multibrot set of power $n$ is locally connected. The multibrot set of power $n$ is defined as the set of all complex parameters $c$ for which the orbit of $0$ under iteration of the map $z \mapsto z^n + c$ does not escape to infinity; equivalently, it is the set of all $c \in \mathbb{C}$ such that for every non-negative integer $k$, the $k$-fold iterate of $z \mapsto z^n + c$ applied to $0$ has modulus at most $2^{(n-1)^{-1}}$ when $n \geq 2$. A topological space is locally connected if every point has a neighborhood base consisting of connected open sets. Prove that for every $n \in \mathbb{N}$, the multibrot set of power $n$, equipped with the subspace topology from $\mathbb{C}$, is a locally connected space.