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Proof

(related to Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition)

Note that \((M, + )\) is not empty, since it contains the Cauchy sequence of rational zeros. It has also already been shown that set \(M\) is closed under the addition of two rational Cauchy sequences. It remains to be shown that \((M, + )\) is a commutative group, which we will do by demonstrating all properties of a commutative group:

1. It has been proven that the addition of rational Cauchy sequences is associative.
2. It has been proven that the addition of rational Cauchy sequences is commutative.
3. It has been shown that the sequence \((0)_{n\in\mathbb N}\) (i.e. consisting of rational zeros only) is a rational Cauchy sequence and neutral with respect to the addition of rational Cauchy sequences.
4. It has been shown that for the rational Cauchy sequence \((a_n)_{n\in\mathbb N}\in M\) the inverse element of the addition "\( + \)" is the rational Cauchy sequence \((-a_n)_{n\in\mathbb N}\in M\).

This completes the proof.

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References

Bibliography

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