(related to Lemma: Unit Ring of All Rational Cauchy Sequences)
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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