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Question about subset

  1. Sep 20, 2012 #1
    Hello, I was looking into this proof

    http://www.proofwiki.org/wiki/Lipschitz_Equivalent_Metrics_are_Topologically_Equivalent

    and I was wondering how they concluded that

    [tex]
    N_{h\epsilon}(f(x);d_2) \subseteq N_{\epsilon}(x;d_1)[/tex]
    [tex]
    N_{\frac{\epsilon}{k}}(f(x);d_1) \subseteq N_{\epsilon}(x;d_2)
    [/tex]

    Couldn't it also be that

    [tex]
    N_{h\epsilon}(f(x);d_2) \supseteq N_{\epsilon}(x;d_1)[/tex]
    [tex]
    N_{\frac{\epsilon}{k}}(f(x);d_1) \supseteq N_{\epsilon}(x;d_2)
    [/tex]


    Thanks!
     
  2. jcsd
  3. Sep 20, 2012 #2

    micromass

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    You have proven that if [itex]y\in N_{h\varepsilon}(f(x);d_2)[/itex], then [itex]y\in N_\varepsilon(x;d_1)[/itex]. This implies that [itex]N_{h\varepsilon}(f(x);d_2)\subseteq N_\varepsilon(x;d_1)[/itex].

    Indeed, saying that [itex]A\subseteq B[/itex] means exactly that all [itex]y\in A[/itex] also have [itex]y\in B[/itex].
     
  4. Sep 20, 2012 #3
    Thanks ;)
     
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