A definition of Cauchy sequence on metric space
Topics
About: metric space
The table of contents of this article
Starting Context
- The reader knows a definition of metric space.
Target Context
- The reader will have a definition of Cauchy sequence on metric space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
For any metric space, \(M\), any sequence, \(s: \mathbb{N} \to M\), where \(\mathbb{N}\) is the positive natural numbers set, such that for any \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\), there is an \(N \in \mathbb{N}\) such that for any \(n_1, n_2 \in \mathbb{N}\) such that \(N \lt n_1, n_2\), \(dist (s (n_1), s (n_2)) \lt \epsilon\)
2: Note
Sometimes, we may let \(\mathbb{N}\) be the natural numbers set (with \(0\)).
