# Proof

We want to prove that the set of all rational Cauchy sequences. $M:=\{(a_n)_{n\in\mathbb N}~|~(a_n)_{n\in\mathbb N}\text{ is a rational Cauchy sequence}\},$ together with the addition and the multiplication of rational Cauchy sequences defined for $$(a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\in M$$ as: $\begin{array}{cclcl} (a_n)_{n\in\mathbb N}+(b_n)_{n\in\mathbb N}&:=&(a_n+b_n)_{n\in\mathbb N}&\quad\quad&\text{"addition"}\\ (a_n)_{n\in\mathbb N}\cdot(b_n)_{n\in\mathbb N}&:=&(a_n\cdot b_n)_{n\in\mathbb N}&\quad\quad&\text{"multiplication"}\\ \end{array}$ forms the commutative unit ring $$(M, + ,\cdot)$$. We prove this lemma in five steps. For simplicity reasons, we write for each rational Cauchy sequence $$(a_n)$$ instead of $$(a_n)_{n\in\mathbb N}$$.

$$(1)$$ It has been demonstrated that $$(M,+)$$ is a commutative group with respect to addition. $$(2)$$ It has been shown that set $$(M,\cdot)$$ is a commutative monoid with respect to multiplication. $$(3)$$ The distributivity laws hold for "$$+$$" and "$$\cdot$$" has been shown in the corresponding proposition. Altogether, we have proven that the resulting algebraic structure $$(M, + ,\cdot)$$ is a commutative unit ring, es required.

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### References

#### Bibliography

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