A convergent rational sequence is a rational sequence \((x_n)_{n\in\mathbb N}\), which is convergent in the metric space of rational numbers \((\mathbb Q,|~|)\).
In other words, \((x_n)_{n\in\mathbb N}\) is convergent to the number \(x\in\mathbb Q\), if for each rational number \(\epsilon > 0\) there exists a natural number \(N\in\mathbb N\) with \[ | x_n-x | < \epsilon\quad\quad \text{ for all }n\ge N.\]
If \((x_n)_{n\in\mathbb N}\) is convergent to the number \(x\in\mathbb Q\), we write \[\lim_{n\to\infty} x_n=x.\]
Lemmas: 1 2 3
Proofs: 4 5 6 7 8 9 10 11 12 13
Propositions: 14 15 16
