(related to Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition)
For every member \(x_n\) of a given rational Cauchy Sequence \((x_n)_{n\in\mathbb N}\), there exists an inverse rational number \(-x_n\) , such that \(x_n + (-x_n)=0\), where \(0\) is the rational zero. By definition of the addition of both sequences we get
\[(x_n)_{n\in\mathbb N}+(-x_n)_{n\in\mathbb N}=(0)_{n\in\mathbb N}.\]
where \((0)_{n\in\mathbb N}\) equals the Cauchy sequence of rational zeros, as required.