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: Structured Description
Here is the rules of Structured Description.
Entities:
\( M\): \(\in \{\text{ the metric spaces }\}\)
\( s\): \(: \mathbb{N} \to M\), \(\in \{\text{ the sequences of points on } M\}\)
\(*m\): \(\in M\)
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Statements:
\(\forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (\exists N \in \mathbb{N} (\forall n \in \mathbb{N} \text{ such that } N \lt n (dist (m, s (n)) \lt \epsilon)))\)
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2: Note
Sometimes, we may use \(\mathbb{N} \setminus \{0\}\) instead of \(\mathbb{N}\).
When \(M\) is made the topological space with the canonical induced topology, \(\mathbb{N}\) is a directed index 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, \(U_m \subseteq M\), of \(m\), there is an \(\epsilon\) open ball, \(B_{m, \epsilon} \subseteq U_m\), around \(m\), and as \(s (n) \in B_{m, \epsilon}\), \(s (n) \in U_m\).
The convergence, \(m\), is inevitably unique: let us suppose that there was another convergence, \(m' \in M\), which implies that \(0 \lt dist (m, m')\); \(dist (m, m') \le dist (m, s (j)) + dist (s (j), m')\), so, \(dist (m, m') - dist (m, s (j)) \le dist (s (j), m')\); for each \(N \lt n\), \(dist (m, s (j)) \lt dist (m, m') / 2\); so, \(dist (m, m') / 2 = dist (m, m') - dist (m, m') / 2 \le dist (s (j), m')\), which means that \(s\) would not converge to \(m'\).
