# Proof

Because the multiplication of rational Cauchy sequences is commutative, it is without loss of generality sufficient to show the right cancellation property, i.e. $\exists N\in\mathbb N:~ (x_n)_{n > N} \cdot (z_n)_{n > N}=(y_n)_{n > N} \cdot (z_n)_{n > N}\Longleftrightarrow (x_n)_{n > N}=(y_n)_{n > N},$ for all rational Cauchy sequences $$(x_n)_{n\in\mathbb N}, (y_n)_{n\in\mathbb N}, (z_n)_{n\in\mathbb N}$$ such that $$(z_n)_{n\in\mathbb N}$$ is not convergent to $$0$$.

Because by hypothesis $$(z_n)_{n\in\mathbb N}$$ is a rational Cauchy sequence not convergent to $$0$$, there exist an index $$N\in\mathbb N$$ such that $$|z_n|>0$$ for all $$n > N$$. For all sequence members with indices $$n > N$$, it follows from the cancellation property of multiplication of rational numbers and from the definition of multiplying rational Cauchy sequences that

$\begin{array}{rcl} (x_n)_{n > N} \cdot (z_n)_{n > N}=(y_n)_{n\in\mathbb N} \cdot (z_n)_{n > N}&\Longleftrightarrow&(x_n\cdot z_n)_{n > N}=(y_n\cdot z_n)_{n > N}\\ &\Longleftrightarrow&(x_n)_{n > N}=(y_n)_{n > N}. \end{array}$

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### References

#### Bibliography

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