(related to Proposition: Distributivity Law For Rational Cauchy Sequences)
Let \((x_n)_{n\in\mathbb N}\), \((y_n)_{n\in\mathbb N}\), and \((z_n)_{n\in\mathbb N}\) be rational Cauchy sequences. Because multiplying rational Cauchy sequences is commutative, it is sufficient to show the left-distributivity law. \[(x_n)_{n\in\mathbb N}\cdot[(y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}]=[(x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}]+[(x_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}].\]
The left-distributivity law can be proven using the following mathematical definitions and concepts: * definition of adding rational Cauchy sequences, * definition of multiplying rational Cauchy sequences, and * distributivity law for rational numbers. The proof follows:
\[\begin{array}{ccll} (x_n)_{n\in\mathbb N}\cdot[(y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}]&=&(x_n)_{n\in\mathbb N}\cdot(y_n+z_n)_{n\in\mathbb N}&\text{by definition of adding rational Cauchy sequences}\\ &=&(x_n\cdot [y_n+z_n])_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&([x_n\cdot y_n]+[x_n\cdot z_n])_{n\in\mathbb N}&\text{by distributivity law of rational numbers}\\ &=&(x_n\cdot y_n)_{n\in\mathbb N}+(x_n\cdot z_n)_{n\in\mathbb N}&\text{by definition of adding rational Cauchy sequences}\\ &=&[(x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}]+[(x_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}]&\text{by definition of multiplying rational Cauchy sequences} \end{array} \]
