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Unformatted text preview: ARE211 FINAL EXAM DECEMBER 12, 2003 This is the final exam for ARE211. As announced earlier, this is an openbook exam. Try to allocate your 180 minutes in this exam wisely, and keep in mind that leaving any questions unanswered is not a good strategy. Make sure that you do all the easy questions, and easy parts of hard questions, before you move onto the hard questions. Problem 1 (15 points) (1) In Euclidean space, under the Pythagorean metric, assume that a sequence ( x n ) satisfies  x n x n +1  ≤ α  x n x n 1  for each n = 2 , 3 ,... for some fixed 0 < α < 1. Show that ( x n ) is a convergent sequence. (2) Let ( X,d ) be a complete metric space. A function f : X → X is called a contraction if there exists some < α < 1 such that ∀ x,y ∈ X, d ( f ( x ) ,f ( y )) ≤ αd ( x,y ) α is called a contraction constant. Show that for every contraction f on a complete metric space ( X,d ), there exists a unique point x ∈ X such that f ( x ) = x . (Such a point x ∈ X is called a fixed point .) (Hint: To prove this, construct a sequence that has the property defined in (1). An understanding of com pleteness will also be helpful.)pleteness will also be helpful....
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Simon

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