# Proof

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

$\begin{array}{ccll} [(x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}]\cdot (z_n)_{n\in\mathbb N}&=&(x_n\cdot y_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&((x_n\cdot y_n)\cdot z_n)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&(x_n\cdot (y_n\cdot z_n))_{n\in\mathbb N}&\text{by associativity of multiplying rational numbers}\\ &=&(x_n)_{n\in\mathbb N}\cdot (y_n\cdot z_n)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&(x_n)_{n\in\mathbb N}\cdot [(y_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}]&\text{by definition of multiplying rational Cauchy sequences}\\ \end{array}$

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013