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Proof

(related to Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences))

• Note that since the rational one \(1\) exists, it is also true that the rational sequence \((1)_{n\in\mathbb N}\) exists, i.e. a sequence whose members all equal the rational one \(1\).
• For any rational \(\epsilon > 0\) we have \[|1_i-1_j|=0 < \epsilon\quad\quad\text{ for all }i,j\ge 1.\]
• Therefore, the sequence \((1)_{n\in\mathbb N}\) is, trivially, a rational Cauchy sequence.
• Let \((x_n)_{n\in\mathbb N}\)be a rational Cauchy sequence.
• It follows from the definition of multiplying rational Cauchy sequences and because rational one is neutral with respect to the multiplication of rational numbers that

\[\begin{array}{ccll} (x_n)_{n\in\mathbb N}\cdot (1)_{n\in\mathbb N}&=&(x_n\cdot 1)_{n\in\mathbb N}&\text{by definition of multiplying rational Cauchy sequences}\\ &=&(x_n)_{n\in\mathbb N}&\text{because }1\text{ is neutral with respect to the multiplication of rational numbers}\\ \end{array}\] * From the commutativity of multiplying rational Cauchy sequences, it follows that also the equation \((1)_{n\in\mathbb N}\cdot (x_n)_{n\in\mathbb N}=(x_n)_{n\in\mathbb N}\) holds for all rational Cauchy sequences \((x_n)_{n\in\mathbb N}\).

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References

Bibliography

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