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

$$N_{h\epsilon}(f(x);d_2) \subseteq N_{\epsilon}(x;d_1)$$
$$N_{\frac{\epsilon}{k}}(f(x);d_1) \subseteq N_{\epsilon}(x;d_2)$$

Couldn't it also be that

$$N_{h\epsilon}(f(x);d_2) \supseteq N_{\epsilon}(x;d_1)$$
$$N_{\frac{\epsilon}{k}}(f(x);d_1) \supseteq N_{\epsilon}(x;d_2)$$


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 ;)