# 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 adding rational Cauchy sequences and from the associativity of adding rational numbers that

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

Github: ### References

#### Bibliography

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