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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 open-book 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