mth536_problem_7_22c

Mth536_problem_7_22c - MTH536 Chapter 7 Problem 22c Gene Quinn – p 1 Chapter 7 Problem 22c If ¢ £ ¤¥ is a metric space show that the metric

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Unformatted text preview: MTH536 Chapter 7 Problem 22c Gene Quinn – p. 1/ Chapter 7 Problem 22c If ¢ £¡ ¤¥ is a metric space, show that the metric is uniformly equivalent to . ¤ ¤ ©¨ ¤ § ¦ – p. 2/ Proof of proposition We must show that given there exists a  such that   ¤      ¦  ¢ and ¦     ¤  ¢    ¢ ¢      – p. 3/ Proof of proposition We must show that given there exists a  such that   ¤      ¦  ¢ and ¦     Let be given and let be arbitrary elements of ¤  ¢    ¢ ¢      . ¡     Choose ¢  !  ©¨  §  ¡ – p. 3/ Proof of proposition We must show that given there exists a  such that   ¤      ¦  ¢ and ¦     Let be given and let be arbitrary elements of ¤  ¢    ¢ ¢      . ¡     Choose ¢  ! ¤     ¤   ©¨  §  First suppose . Then since  #" and ,  ¢ © ¨ ¤  ¤ ¢ ¢   $ ¤  ¢   ©¨      ©¨  ¤  ¢ §    ¢    ¡ – p. 3/ which implies that Now suppose that Proof of proposition ¢    ¦  ¢ ¦     . Then ©¨ ¤  ¢  § ¢ ¢ ¦     © ¨ ¤  ¤ ¢ ¢    ©¨   ¤   – p. 4/ Proof of proposition which implies that ¤  ¢ ¦  § ¦  Now suppose that    ¢  Let be given and let © ¨ ¤  ¤ ¢ ¢    ©¨   be arbitrary elements of ¦  ¢    ©¨ ¤  ¢  . Then ¢    . ¡     Choose ¢  !  ©¨  §  ¡ – p. 4/ First suppose ¢ §  ¤     ©¨  Proof of proposition ¢ ¤     . Then since ¢ $ ¤ ¢    ©¨    ¤   ©¨ ¤  ¢  ¢ ¤   #"  and    , – p. 5/ ¢ First suppose  ! ¡ ¢  © ¢ ¤     ¢ ¦  ¢  ¦     . Then Since  ¢ § ¤     ©¨  Proof of proposition Now suppose that and .  ¤  ¢ ¤     . Then since were arbitrary elements of ¢ ¡ $ ¤ ¢ ¢  ¤       ©¨    ¤    %&    © ¤  ¢ %   ©¨ ¤  ¢  ¢ ¤   #"  and © ¨ ¤ ¤  ¢  ¢    ©¨   , the result holds for all    , – p. 5/ ...
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This note was uploaded on 03/23/2010 for the course MATH 515 taught by Professor Staff during the Spring '08 term at Iowa State.

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