Let \((X,|~|)\) be a mectric space and let \((a_n)_{n\in\mathbb N}\) be a sequence of points in \(X\). The sequence \((a_n)\) is called Cauchy sequence, if for all \(\epsilon > 0\) there exists an \(N(\epsilon)\in\mathbb N\) with \[|a_i-a_j| < \epsilon\quad\quad\text{ for all }i,j\ge N(\epsilon).\]
\(N(\epsilon)\) means that the natural number \(N\) depends only on \(\epsilon\).
Definitions: 1 2
Lemmas: 3
Proofs: 4 5 6 7
Propositions: 8