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mth536_problem_7_9a

# mth536_problem_7_9a - MTH536 Chapter 7 Problem 9a Gene...

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Unformatted text preview: MTH536 Chapter 7 Problem 9a Gene Quinn – p. 1/ Chapter 7 Problem 9a Let be a point and a set in a metric space. Deﬁne £ ¡¦ £ ¤ ¥ © ¨ ¢  ¢ §    ¤ £ ¦ ¦ ¡ £ Show that for a ﬁxed continuous. , the function given by ¡¦ ¢ ¤ ¥ § is ¡  – p. 2/ Proof of proposition We need to show that, for any given ! ¦ £ ¦ ! £   # "  , there is a £ ¦ ¢ ¤ such that   whenever  #   – p. 3/ Proof of proposition We need to show that, for any given ! ¦ £ ¦ ! £   # "  , there is a £ ¦ ¢ ¤ such that   whenever  Let be given. Now ¦ £ £ ¡¦  ¤ ¥ § § © ¨ ¢ \$    so there must exist a    with %¦ £ ¡¦ ¢ ¤ ¥ "  # '( % ¢ for otherwise, we would have £ %¦ ¢£ ¡¦ 1 ) 0 ¢ ¤ ¤  % & ¡ '( ¤ ¥ £ & ¡ £ contradicting the deﬁnition of ¢ ¤ ¡¦ . #   – p. 3/ Proof of proposition or ¤ ¢£ ¡¦ ¢ ¤ ¤ ¡¦ " ¢ ¤ ¡¦ ¢ ¤ ¥ ¦  0 '( ¢ ¤ ¥ ¡¦  0 '( # £ ¢ ¤ ¦ 0 £ £ ¢ ¤ ¥ %¦ # £ ¢ £ ¢ ¤ ¦ 0 £ %¦ 2 £ 2 £ % Then for this we have – p. 4/ Proof of proposition or ¢ ¤ ¥ ¤ ¡¦ ¢ ¤ ¥ ¢ ¤ ¥ ¡¦ ¢ ¤ ¥ £ 0 ¡¦ ¢ ¤ ¥ ¦  0 '( ¢ ¤ ¡¦  0 '(  0 '( £ # ¤ 3 ¦ 2 ¤ ¢£ ¦ 0 £ ¢ ¤ ¥ ¡¦  # '(  ¢£ ¦ " ¤ ¡¦ ¢ ¤ ¥ ¦ 0 '( ¢ £ £ ¤ 3 ¦ £ " ¤ ¢ 2 ¢ £ ¦ 3 & ¡ " ¢ ¤ £ 3 £ ¢ ¤ ¡¦ £ ¢ ¤ ¦ 0 2 3 £ ¡¦ ¢ ¤ %¦ 2 £ 2 £ ¢£ % or and for this we have we can write with ¡¦ # £ ¢ ¤ ¦ 0 £ ¢ ¤ ¥ %¦ # £ £ Then for this Similarly there must be a – p. 4/ Proof of proposition Combining inequalities gives  £ ¦ ¢£ ¡¦ £ ¡¦ £ ¦ ¤ ¥ £ ¢ ¤ # ¢ ¤ ¥ ¤ ¢ ¤ ¥ (" " " ¢ # ( 0 ( 0  ! £ ¡¦ £ ¡¦ ! ! ¦ £ ¦ ! £ ¤ ¤ " §  which is the same as " ¦   ¢ ¢ # which holds whenever £ ¦ ¢ ¤ ¥ ( #  – p. 5/ ...
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