Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative

The multiplication of rational Cauchy sequences is cancellative, i.e. for all rational Cauchy sequences \((x_n)_{n\in\mathbb N}\), \((y_n)_{n\in\mathbb N}\) and \((z_n)_{n\in\mathbb N}\) such that \((z_n)_{n\in\mathbb N}\) is not convergent to \(0\), there exists \(N\in\mathbb N\) such that for all sequence members with indices \(n > N\), following laws (both) are fulfilled:

Conversely, for a rational Cauchy sequence \((z_n)_{n\in\mathbb N}\), which is not convergent to \(0\), there exists a natural number \(N\) such that the equation \((x_n)_{n > N}=(y_n)_{n > N}\) implies * \((z_n)_{n > N}\cdot (x_n)_{n > N}=(z_n)_{n > N} \cdot (y_n)_{n > N}\) and * \((x_n)_{n > N}\cdot (z_n)_{n > N}=(x_n)_{n > N} \cdot (z_n)_{n > N}\).

Proofs: 1

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