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Math Help - Help with fractals

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    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)
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    Quote Originally Posted by SamBourne View Post
    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_ms, 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.
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