204Section42009 - Econ 204 Section 4 GSI: Hui Zheng Key...

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Econ 204 Section 4 GSI: Hui Zheng Key Words Metric Space, Normed Vector Space, Euclidean Space, Lipschitz-Equivalent, Convergence, Cluster Point, Increasing(Decreasing) Sequence, Lim Sups(Lim Infs), Rising Sun Lemma, Bolzano-Weierstrass Theorem Section 4.1 Metric Space metric space is a pair ( X;d ) , where X is a set and d : X ± X ! R + , satisfying 1. 8 x;y 2 X d ( x;y ) ² 0 ;d ( x;y ) = 0 , x = y 2. 8 x;y 2 X d ( x;y ) = d ( y;x ) 3. (triangle inequality) 8 x;y;z 2 X d ( x;y ) + d ( y;z ) ² d ( x;z ) Example 4.1.1 Let d ( x;y ) = max f x ³ y ; 1 g . Prove or disprove that ( R ;d ) is a metric space. Disproof: Let x 2 X . Then d ( x;x ) = max f x ³ x ; 1 g = max f 0 ; 1 g = 1 . So d is not a metric. Example 4.1.2 Let d ( x;y ) = min f x ³ y ; 1 g . Prove or disprove that ( R ;d ) is a metric space. Proof:: In fact this is called the standard bounded metric corresponding to d . Check the triangle inequality: d ( x;z ) ´ d ( x;y ) + d ( y;z ) Now if either x ³ y ² 1 or y ³ z ² 1 then the right side of this inequality is at least 1 ; 1 , the inequality holds. It remains to consider the case in which x ³ y < 1 and y ³ z < 1 . In this case, we have x ³ z ´ x ³ y + y ³ z = d ( x;y ) + d ( y;z ) : Hence d ( x;z ) = min f x ³ z ; 1 g ´ x ³ z ´ d ( x;y ) + d ( y;z ) : The triangle inequality holds. Example 4.1.3 Let X = [1 ; + 1 ) . Let d ( x;y ) = 1 x ³ 1 y . Prove or disprove that ( X;d ) is a metric space. Proof: 8 x;y 2 X; d ( x;y ) = 1 x ³ 1 y ² 0 and d ( x;y ) = 1 x ³ 1 y = 0 , x = y 8 x;y 2 X; d ( x;y ) = 1 x ³ 1 y = 1 y ³ 1 x = d ( y;x ) Check the triangle inequality. We show that d ( x;z ) ´ d ( x;y ) + d ( y;z ) will depend upon the ordering of x , y , and z . Because d ( x;z ) = d ( z;x ) , without loss of generality, we can assume x ´ z . Case 1. Suppose 1 x ² 1 y ² 1 z . Then d ( x;y ) + d ( y;z ) = 1 x ³ 1 y + 1 y ³ 1 z = 1 x ³ 1 y + 1 y ³ 1 z = 1 x ³ 1 z = 1 x ³ 1 z = d ( x;z ) Case 2. Suppose 1 x ² 1 z ² 1 y . Then d ( x;y ) + d ( y;z ) = 1 x ³ 1 y + 1 y ³ 1 z = 1 x ³ 1 y + 1 z ³ 1 y =
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204Section42009 - Econ 204 Section 4 GSI: Hui Zheng Key...

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