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Unformatted text preview: ance is a metric on K(R2 ), the set of all compact
subsets of the plane (Prop. 2.2.1).
2. Show T is an h-contraction if each Ti is a d-contraction (Prop. 2.3.2).
3. Given an IFS T , show there is a unique compact set A for which
T (A) = A (Theorem 2.3.1(a)).
4. For any compact set B , T n (B ) → A as n → ∞, in the sense that
limn→∞ h(T n (B ), A) = 0 (Theorem 2.3.1(b)).
The ﬁrst step, the step we take in this section, is showing h is a metric. The
remaining steps are in Sect. 2.3.
Despite the brevity of the proofs in this section, here we begin a method of
presenting arguments that we follow for the rest of this text. To make more
transparent the logical ﬂow of arguments, we present statements of lemmas,
propositions, theorems, and corollaries, together with their connective material ﬁrst, then all the proofs afterward, just before the practice problems. This
ordering violates several millenia of mathematica practice; it was inspired by experience writing webpages, where each lemma, etc., is...
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