Let \((\mathbb Q,|~|)\) be the mectric space of all rational numbers, together with the distance defined by the absolute value "\(|~|\)", and let \((a_n)_{n\in\mathbb N}\) be a sequence of rational numbers in \((\mathbb Q,|~|)\). The sequence \((a_n)\) is called a rational Cauchy sequence, if for all rational numbers \(\epsilon > 0\) there exists an index1 \(N(\epsilon)\in\mathbb N\) with \[|a_i-a_j| < \epsilon\quad\quad\text{ for all }i,j\ge N(\epsilon).\]
Definitions: 1 2 3
Lemmas: 4 5 6
Parts: 7
Proofs: 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Propositions: 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
\(N(\epsilon)\) means that the natural number \(N\) depends only on the rational number \(\epsilon\). ↩
