(related to Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication)
Note that \((M, \cdot )\) is not empty, since it contains the Cauchy sequence of rational ones. It has also already been shown that set \(M\) is closed under the multiplication of two rational Cauchy sequences. It remains to be shown that \((M, + )\) is a commutative monoid, which we will do by demonstrating all properties of a commutative monoid:
\((i)\) It has been proven that the multiplication of rational Cauchy sequences is associative. \((ii)\) It has been shown that the multiplication of rational Cauchy sequences is commutative. \((iii)\) It has been shown that the sequence \((1)_{n\in\mathbb N}\) (i.e. consisting of rational ones only) is a rational Cauchy sequence and neutral with respect to the multiplication of rational Cauchy sequences. This completes the proof.
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