I want to give an example of a Cauchy sequence of positive rational numbers that is equivalent to a Cauchy sequence of negative rational numbers - if possible. Here is what I did:
So, two Cauchy sequences of rational numbers $\{a_n\}_n$ and $\{b_n\}_n$ are equivalent if and only if $lim_{n\rightarrow\infty} a_n - b_n = 0$. In other words, if and only if $\{a_n - b_n\}_n$ converges to zero.
Let $a_n = \frac{1}{n}$ and $b_n = \frac{-1}{n}$.
Let $\epsilon\in\mathbb{Q}^+$ and choose $N > \frac{2}{\epsilon}$, which may be done by the archimedian property. It follows that:
$|a_n - b_n | = |\frac{1}{n}-(\frac{-1}{n})| = |\frac{2}{n}| = \frac{2}{n} < \epsilon$, because $n \geq N > \frac{2}{\epsilon}$ implies $\epsilon > \frac{2}{n}$.
Thus $\{a_n-b_n\}_n$ converges to zero and $\{\frac{1}{n}\}_n\sim\{\frac{-1}{n}\}_n$. So a Cauchy sequence of positive rational numbers CAN be equivalent to a Cauchy sequence of negative rational numbers.
Is this correct? Basically, they're equivalent because they're converging to the same value? Because the sequences are eventually epsilon-close?