(related to Proposition: Addition of Rational Cauchy Sequences Is Commutative)
Let \((x_n)_{n\in\mathbb N}\) and \((y_n)_{n\in\mathbb N}\) be rational Cauchy sequences. It follows from the definition of adding rational Cauchy sequences and from the commutativity of adding rational numbers that
\[\begin{array}{ccll} (x_n)_{n\in\mathbb N}+(y_n)_{n\in\mathbb N}&=&(x_n+y_n)_{n\in\mathbb N}&\text{by definition of adding rational Cauchy sequences}\\ &=&(y_n+x_n)_{n\in\mathbb N}&\text{by commutativity of adding rational numbers}\\ &=&(y_n)_{n\in\mathbb N}+(x_n)_{n\in\mathbb N}&\text{by definition of adding rational Cauchy sequences}\\ \end{array}\]
