1. Sep 20, 2012 #1

   PhysicsGente

   

   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!

    

   PhysicsGente, Sep 20, 2012
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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].

    

   micromass, Sep 20, 2012
4. Sep 20, 2012 #3

   PhysicsGente

   

   Thanks ;)

    

   PhysicsGente, Sep 20, 2012

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