mth536_problem_7_9b

mth536_problem_7_9b - MTH536 Chapter 7 Problem 9b Gene...

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Unformatted text preview: MTH536 Chapter 7 Problem 9b Gene Quinn – p. 1/ Chapter 7 Problem 9b Let be a point and a set in a metric space. Define £ ¡¦ £ ¤ ¥ © ¨ ¢  ¢ §    ¤   ¢ ¤ ¥ §  § ¡ ¦ ¡ Show that  £ ¡¦ – p. 2/ Proof of proposition Let . Suppose that £ ¦ ©¨  ! § "  ¢    ¤ ¥ ¡ £ Then by definition for any contradicting that there exists a £ ¤ ¥ ¦ ©¨  ¢ with #¦ ¢ ¤ , ! "  ¡ ! § so it must be that £ ¦ © ¨  ¢ ¤    §     # $ ! – p. 3/ Proof of proposition Since was an arbitrary choice, it must be true that £ ¡¦ %  £ § ¡¦ ¡ &  ¢ ¤ ¥ §  ¡ ¢ ¤ ¥ and we conclude that   – p. 4/ Proof of proposition £ Let ¡¦ and let be given. Now £ ¦ ¢ ¤ ¥ £    § #¦ ¢ ¡  ¤ ¥  § &  ! "  ¢ ¤ ¥ §  implies that there exists a It follows that  ©¨ with ¡ . ¡ is a limit point of # and therefore and we conclude that  ¢£ ¡¦  ¤ ¡ $ ! – p. 5/ Proof of proposition £ Let ¡¦ and let be given. Now £ ¦ ¢ ¤ ¥ £    § #¦ ¢ ¡  ¤ ¥  §  & ¡¦ §  § ¡  ! "  ¢ ¤ ¥ §  implies that there exists a It follows that  ©¨ with ¡ . ¡ is a limit point of # and therefore and we conclude that  ¢£ ¡¦  ¤ ¡ Together with the previous result this establishes that   ¤ ¢£ $ ! – p. 5/ ...
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