Define the Hausdorff dimension of a subset of the Euclidean plane.

Let $\Delta$ be a closed disc of radius $r_{0}$ in the Euclidean plane. Define a sequence of sets $K_{n} \subseteq \Delta, n=1,2, \ldots$, as follows: $K_{1}=\Delta$ and for each $n \geqslant 1$ a subset $K_{n+1} \subset K_{n}$ is produced by replacing each component disc $\Gamma$ of $K_{n}$ by three disjoint, closed discs inside $\Gamma$ with radius at most $c_{n}$ times the radius of $\Gamma$. Let $K$ be the intersection of the sets $K_{n}$. Show that if the factors $c_{n}$ converge to a limit $c$ with $0<c<1$, then the Hausdorff dimension of $K$ is at most $\log \frac{1}{3} / \log c$.