(related to Proposition: Multiplication 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 multiplying rational Cauchy sequences and from the associativity of multiplying rational numbers that
\[\begin{array}{ccll} [(x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}]\cdot (z_n)_{n\in\mathbb N}&=&(x_n\cdot y_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&((x_n\cdot y_n)\cdot z_n)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&(x_n\cdot (y_n\cdot z_n))_{n\in\mathbb N}&\text{by associativity of multiplying rational numbers}\\ &=&(x_n)_{n\in\mathbb N}\cdot (y_n\cdot z_n)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&(x_n)_{n\in\mathbb N}\cdot [(y_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}]&\text{by definition of multiplying rational Cauchy sequences}\\ \end{array}\]
