Proposition: Multiplication Of Rational Cauchy Sequences

Let $(x_n)_{n\in\mathbb N}$ and $(y_n)_{n\in\mathbb N}$ be two rational Cauchy sequences. The multiplication of rational Cauchy Sequences can be defined based on the multiplication of rational numbers as follows:

$\begin{array}{cclc} (x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}&:=&(x_n\cdot y_n)_{n\in\mathbb N}\\ \end{array}$ i.e. the product of two rational Cauchy sequences is a new rational sequence, whose sequence members are the products of rational numbers being the corresponding members of the original two rational Cauchy sequences.

Moreover, the new rational sequence $(x_n\cdot y_n)_{n\in\mathbb N}$ is also a rational Cauchy sequence.

Proofs: 1