# 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 $$(x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N},(z_n)_{n\in\mathbb N}$$.

Proofs: 1

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