Lemma: Unit Ring of All Rational Cauchy Sequences

The set of all rational Cauchy sequences. \[M:=\{(a_n)_{n\in\mathbb N}~|~(a_n)_{n\in\mathbb N}\text{ is a rational Cauchy sequence}\},\]

together with the two operations defined for any two rational Cauchy sequences \((a_n)_{n\in\mathbb N},(b_n)_{n\in\mathbb N}\in M\) as follows: \[\begin{array}{cclcl} (a_n)_{n\in\mathbb N}+(b_n)_{n\in\mathbb N}&:=&(a_n+b_n)_{n\in\mathbb N}&\quad\quad&\text{"addition"}\\ (a_n)_{n\in\mathbb N}\cdot(b_n)_{n\in\mathbb N}&:=&(a_n\cdot b_n)_{n\in\mathbb N}&\quad\quad&\text{"multiplication"}\\ \end{array}\]

builds an algebraic structure \((M, + ,\cdot)\), which is a commutative unit ring.

Proofs: 1

Lemmas: 1 2
Proofs: 3 4 5


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References

Bibliography

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