(related to Proposition: Addition of Rational Cauchy Sequences Is Cancellative)

Because the addition of rational Cauchy sequences is commutative, it is without loss of generality sufficient to show the right cancellation property, i.e. \[(x_n)_{n\in\mathbb N} + (z_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N} + (z_n)_{n\in\mathbb N}\Longleftrightarrow (x_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N},\] for all rational Cauchy sequences \((x_n)_{n\in\mathbb N}, (y_n)_{n\in\mathbb N}, (z_n)_{n\in\mathbb N}\). It follows from the definition of adding rational Cauchy sequences, and because the addition of rational numbers is cancellative:

\[\begin{array}{rcll} (x_n)_{n\in\mathbb N} + (z_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N} + (z_n)_{n\in\mathbb N}&\Longleftrightarrow& (x_n+z_n)_{n\in\mathbb N}=(y_n+z_n)_{n\in\mathbb N}\\ &\Longleftrightarrow& (x_n)_{n\in\mathbb N}=(y_n)_{n\in\mathbb N}\\ \end{array}\]

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