Understanding Subset Equivalence in Lipschitz Equivalent Metrics Proof

[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

<br /> N_{h\epsilon}(f(x);d_2) \subseteq N_{\epsilon}(x;d_1)
<br /> N_{\frac{\epsilon}{k}}(f(x);d_1) \subseteq N_{\epsilon}(x;d_2)<br />

Couldn't it also be that

<br /> N_{h\epsilon}(f(x);d_2) \supseteq N_{\epsilon}(x;d_1)
<br /> N_{\frac{\epsilon}{k}}(f(x);d_1) \supseteq N_{\epsilon}(x;d_2)<br />


Thanks!

 

[micromass]

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

Indeed, saying that A\subseteq B means exactly that all y\in A also have y\in B.

 

[PhysicsGente]

Thanks ;)