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