I have two metrics on $\mathbb R$: $d_1(x,y)=|x-y|,$ $d_2(x,y)=|x^3-y^3|.$
And I want to show that the uniform structures $\mathcal{U}_{d_1}$ and $\mathcal{U}_{d_2}$ defined by the respective bases $U_{\epsilon_n}=\{(x,y) : d_n(x,y)<\epsilon_n\}$ for $n=1,2$ are not equivalent.
What I showed is that if we take the same epsilon, then there can be different basis elements, which obviously are in their respective uniform structure but not in the other.
For example, take $\epsilon = 2$, and take the point $(1,2)$, in the basis element with the metric of powers of $3$, this point isn't in it, but in the basis element with the usual metric the point belongs to it, so we have two different basis elements for each uniformity, so we must have one set that is in one uniformity but not in the other.
Is this argument valid?
Thanks.