(related to Lemma: Convergent Rational Sequences With Limit \(0\) Are an Ideal Of the Ring of Rational Cauchy Sequences)

We want to show that the set \(I:=\{(a_n)_{n\in\mathbb N}~|~a_n\in\mathbb Q,\lim a_n=0\}\) of all rational sequences convergent to \(0\) is an ideal of the commutative unit ring of all rational Cauchy sequences \((M , + , \cdot)\), formally \(I\lhd R\).

We will do so by proving all properties of an ideal:

\((1)\) It has been proven already that \((I, +)\) is a subgroup of \((M , +)\).

\((2)\) We want to show that \((c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (c_n\cdot a_n)_{n\in\mathbb N}\in I.\)

It has been shown that given a rational Cauchy sequence \((c_n)_{n\in\mathbb N}\), the members \(c_n\) are bounded, i.e. there exists a positive constant \(c > 0, c\in\mathbb Q\), such that \(|c_n|\le c\) for all \(n\in\mathbb N\). By hypothesis, \((a_n)\in I\) is a rational sequence convergent to \(0\). Thus, we can choose any arbitrarily small \(\epsilon\in\mathbb Q,\epsilon > 0\), and conclude that there exists an \(N(\epsilon/c)\in\mathbb N\) such that for all \(n > N(\epsilon/c)\) we have the estimation \[|c_n\cdot a_n|=|c_n|\cdot|a_n|\le c\cdot \frac\epsilon c=\epsilon.\] This shows that a right-product of a rational Cauchy sequence \((c_n)_{n\in\mathbb N}\) and a rational sequence \((a_n)_{n\in\mathbb N}\) convergent to \(0\) is a sequence \((c_n\cdot a_n)_{n\in\mathbb N}\), which also converges to \(0\), or equivalently an element of \(I\).

\((3)\) We want to show that \((c_n)_{n\in\mathbb N}\in M,~(a_n)_{n\in\mathbb N}\in I\Longrightarrow (a_n\cdot c_n)_{n\in\mathbb N}\in I.\)

This can be concluded immediately from \((2)\), since \((M, + ,\cdot)\) is a commutative ring by hypothesis.

This demonstrates altogether that \(I\) is an ideal of \((M, + ,\cdot)\).

Thank you to the contributors under CC BY-SA 4.0!




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