Question about subset

  • #1

Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
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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].
 
  • #3
Thanks ;)
 

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