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Proposition: Addition of Rational Cauchy Sequences Is Cancellative

The addition of rational Cauchy sequences is cancellative, i.e. for all rational Cauchy sequences \((x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N},(z_n)_{n\in\mathbb N}\), the following laws (both) are fulfilled:

• Left cancellation property: If the equation \((z_n)_{n\in\mathbb N} + (x_n)_{n\in\mathbb N}=(z_n)_{n\in\mathbb N} + (y_n)_{n\in\mathbb N}\) holds, then it implies \((x_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N}\).

• Right cancellation property: If the equation \((x_n)_{n\in\mathbb N} + (z_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N} + (z_n)_{n\in\mathbb N}\) holds, then it implies \((x_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N}\).

Conversely, the equation \((x_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N}\) implies

• \((x_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}\) and
• \((z_n)_{n\in\mathbb N} + (x_n)_{n\in\mathbb N}=(z_n)_{n\in\mathbb N} + (y_n)_{n\in\mathbb N}\)

for all rational Cauchy sequences \((x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N},(z_n)_{n\in\mathbb N}\).

Table of Contents

Proofs: 1

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Proofs: 1 2

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References

Bibliography

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