Note that since the rational zero \(0\) exists , it is also true that the rational sequence \((0)_{n\in\mathbb N}\) exists. We have to show that \((0)_{n\in\mathbb N}\) is a rational Cauchy sequence. But this is trivially the case, since for any rational \(\epsilon > 0\) we have
\[|0_i-0_j|=0 < \epsilon\quad\quad\text{ for all }i,j\ge 1.\]
It remains to be shown that the rational Cauchy sequence \((0)_{n\in\mathbb N}\) is neutral with respect to the addition of rational Cauchy sequences. Let \((x_n)_{n\in\mathbb N}\) be a rational Cauchy sequence. Because \(0\) is neutral with respect to the addition of rational numbers, it follows that
\[\begin{array}{ccll} (x_n)_{n\in\mathbb N}+(0)_{n\in\mathbb N}&=&(x_n+0)_{n\in\mathbb N}&\text{by definition of adding rational Cauchy sequences}\\ &=&(x_n)_{n\in\mathbb N}&\text{because }0\text{ is neutral with respect to the addition of rational numbers}\\ \end{array}\]
It remains to be shown that also the equation \((0)_{n\in\mathbb N}+(x_n)_{n\in\mathbb N}=(x_n)_{n\in\mathbb N}\) holds for all rational Cauchy sequences \((x_n)_{n\in\mathbb N}\). It follows immediately from the commutativity of adding rational Cauchy sequenses.
