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Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication

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 an addition operation defined for any rational Cauchy sequences \((a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\in M\) by: \[\begin{array}{rcl} (a_n)_{n\in\mathbb N}\cdot (b_n)_{n\in\mathbb N}&:=&(a_n\cdot b_n)_{n\in\mathbb N}\\ \end{array}\]

builds an algebraic structure \((M, \cdot)\), which is a commutative monoid.

Table of Contents

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References

Bibliography

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