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
