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Unformatted text preview: 2) Let ( X , d ) be a complete metric space and T be a contraction mapping on X with modulus β ∈ (0, 1). Let x * be the unique fixed point of T . Let x ∈ X , and let n ∈ ϖ . Then by the Triangle Inequality, *). , ( ) , ( *) , ( 1 1 x x T d x T x T d x x T d n n n n + + + ≤ Since T is a contraction, we also have *). , ( *) , ( *) , ( 1 1 x x T d Tx x T d x x T d n n n ≤ = + + Thus, ). , ( 1 1 *) , ( ) , ( *) , ( ) 1 ( *). , ( ) , ( *) , ( 1 1 1 x T x T d x x T d x T x T d x x T d x x T d x T x T d x x T d n n n n n n n n n n + + +≤ ≤+ ≤...
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 Spring '10
 Feigenbaum
 Macroeconomics, Metric space, x*, continuously differentiable function

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