I am reading several books on Fractals and their geometry. Pretty much all of them say "Hutchinson showed you can use the self-similarity property of fractals to help calculate their Hausdorff dimension". None of them, however, give a proof - not even a sketch of one - in their books (in fact, they don't even state it formally). I was hoping someone could (a) show me how to prove it, or (b) explain to me that it is far too difficult to bother with.
Theorem (how I think it should be stated formally). Let $F\subset\mathbb{R}^{m}$ and let $K_{1},\ldots, K_{m}$ be contractions on $\mathbb{R}^{m}$ with Lipschitz constants $c_{1},\ldots,c_{m}$, and let $c=\max\{c_{1},\ldots,c_{m}\}$. Define the Hutchinson operator $K$ on $\mathbb{R}^{m}$ by $K(F)=K_{1}(F)\cup\ldots\cup K_{m}(F)$. Then $K$ is a contraction with Lipschitz constant $c$.
Proof. Not a clue.
Note that Lipschitz mappings are probably not good enough in this context---your quoted text requires that the mappings be self-similar, which is a stronger condition. "Hutchinson's theorem," which deals with self-similar sets, is proved in the paper
This paper is quite approachable, and sets up much of the modern theory of fractal analysis, and the proofs are not that hard to follow. I really wish that someone had brought it to my attention when I was first learning about the Hausdorff measure of self-similar sets.