A definition of convergence of 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 convergence of 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, \(M\), and any sequence, \(s: \mathbb{N} \to M\), where \(\mathbb{N}\) is the positive natural numbers set, a point, \(p \in M\), such that for any \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\), there is an \(N \in \mathbb{N}\) such that for any \(n \in \mathbb{N}\) such that \(N \lt n\), \(dist (p, s (n)) \lt \epsilon\)
2: Note
Sometimes, we may let \(\mathbb{N}\) be the natural numbers set (with \(0\)).
When \(M\) is made the topological space with the canonical induced topology, \(\mathbb{N}\) is a directed set, \(s\) is a net with the directed index set, and any convergence of \(s\) as the sequence is a convergence of \(s\) as the net with the directed index set, because for any open neighborhood, \(U_p \subseteq M\), of \(p\), there is an \(\epsilon\) open ball, \(B_{p, \epsilon} \subseteq U_p\), around \(p\), and as \(s (n) \in B_{p, \epsilon}\), \(s (n) \in U_p\).
