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Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition

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: \[\begin{array}{rcl} (a_n)_{n\in\mathbb N}+(b_n)_{n\in\mathbb N}&:=&(a_n+b_n)_{n\in\mathbb N}\\ \end{array}\]

builds an algebraic structure \((M, +)\), which is a commutative group.

Table of Contents

Proofs: 1

Mentioned in:

Lemmas: 1
Proofs: 2 3 4

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References

Bibliography

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