applicability: $\mathbb {Q, R}$
Let \((\mathbb R,|~|)\) be the metric space of all real numbers, together with the distance defined by the absolute value "\(|~|\)", and let \((a_n)_{n\in\mathbb N}\) be a sequence of real numbers in \((\mathbb R,|~|)\). The sequence \((a_n)\) is called a real Cauchy sequence, if for all \(\epsilon > 0\) there exists an index^{1} \(N(\epsilon)\in\mathbb N\) with \[|a_i-a_j| < \epsilon\quad\quad\text{ for all }i,j\ge N(\epsilon).\]
Corollaries: 1
Proofs: 2 3 4 5 6 7 8
Propositions: 9 10 11
Theorems: 12
\(N(\epsilon)\) means that the natural number \(N\) depends only on the rational number \(\epsilon\). ↩