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