# Proposition: Distributivity Law For Rational Cauchy Sequences

For arbitrary rational Cauchy sequences $$(x_n)_{n\in\mathbb N}$$, $$(y_n)_{n\in\mathbb N}$$, and $$(z_n)_{n\in\mathbb N}$$ with the binary operations addition "$$+$$" and multiplication "$$\cdot$$", the following distributivity laws hold:

"left-distributivity property": $$(x_n)_{n\in\mathbb N}\cdot\lbrack (y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}\rbrack =\lbrack (x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N}\rbrack +\lbrack (x_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N}\rbrack,$$ "right-distributivity property": $$\lbrack (y_n)_{n\in\mathbb N}+(z_n)_{n\in\mathbb N}\rbrack \cdot (x_n)_{n\in\mathbb N}=\lbrack (y_n)_{n\in\mathbb N}\cdot (x_n)_{n\in\mathbb N}]+\lbrack (z_n)_{n\in\mathbb N}\cdot (x_n)_{n\in\mathbb N}\rbrack.$$

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