1. ## Help with fractals

If you have a decreasing sequence of compact subsets of X such that C1⊃C2⊃... Cm and C=∞,m=1, then show that the Hausdorff metric, D(Cn,C)→0 as n→∞.

Assume C∈S(X)

2. Originally Posted by SamBourne
If you have a decreasing sequence of compact subsets of X such that C1⊃C2⊃... Cm and C=∞,m=1, then show that the Hausdorff metric, D(Cn,C)→0 as n→∞.
Since C is the intersection of the $C_m$s, it is sufficient to find a value of m for which every element of $C_m$ is within distance $\varepsilon$ of C (for some given $\varepsilon>0$).

Let $C_\varepsilon = \{x\in C_1:d(x,C)\geqslant\varepsilon\}$. This is a closed (and therefore compact) subset of $C_1$. It is covered by the sets $U_n = \{x\in C_1:x\notin C_n\}\ (n\geqslant1)$, which are open in $C_1$. By compactness there is a finite subcover, and since the sets $U_n$ form an increasing nest, there is in fact just one of them, say $U_m$, that contains $C_\varepsilon$. It follows by taking complements that $C_m\subseteq C_\varepsilon$. Thus $d(C_m,C)<\varepsilon$.