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 sequence.
- 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:
\( J\): \(\subseteq \mathbb{N}\), such that \(J \neq \emptyset\)
\( M\): \(\in \{\text{ the metric spaces }\}\)
\( s\): \(: J \to M\), \(\in \{\text{ the sequences of points on } M\}\)
\(*m\): \(\in M\)
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Statements:
\(\vert J \vert \lt \infty \implies s_{\vert J \vert} = m\)
\(\land\)
\(\vert J \vert = \infty \implies \forall \epsilon \in \mathbb{R} \text{ such that } 0 \lt \epsilon (\exists N \in \mathbb{N} \setminus \{0\} (\forall n \in \mathbb{N} \setminus \{0\} \text{ such that } N \lt n (dist (m, s_n) \lt \epsilon)))\)
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2: Note
Usually, 'convergence' is talked about when \(J\) is infinite, because when \(J\) is finite, there is no necessity for special considerations, although this definition makes the definition also for the \(J\)-finite case, for the completeness of the definition.
When \(M\) is made the topological space with the topology induced by the metric, \(J\) is a directed index set, \(s\) is a net with the directed index set, and any convergence of \(s\) as the sequence on the metric space is a convergence of \(s\) as the net with the directed index set, because (for the \(J\) infinite case, while the \(J\) finite case is obvious) for each open neighborhood, \(U_m \subseteq M\), of \(m\), there is an \(\epsilon\)-'open ball' around \(m\), \(B_{m, \epsilon} \subseteq U_m\), and there is an \(N\) such that for each \(N \lt n\), \(dist (m, s_n) \lt \epsilon\), which means that \(s_n \in B_{m, \epsilon} \subseteq U_m\); on the other hand, any convergence of \(s\) as the net with the directed index set is a convergence of \(s\) as the sequence on the metric space, because for any \(\epsilon\), as \(B_{m, \epsilon}\) is an open neighborhood of \(m\), there is an \(N\) such that for each \(N \lt n\), \(s_n \in B_{m, \epsilon}\), which means that \(dist (m, s_n) \lt \epsilon\).
The convergence, \(m\), is inevitably unique: let us suppose that there was another convergence, \(m' \in M\), which would imply that \(0 \lt dist (m, m')\); \(dist (m, m') \le dist (m, s_n) + dist (s_n, m')\), so, \(dist (m, m') - dist (m, s_n) \le dist (s_n, m')\); for each \(N \lt n\), \(dist (m, s_n) \lt dist (m, m') / 2\); so, \(dist (m, m') / 2 = dist (m, m') - dist (m, m') / 2 \lt dist (s_n, m')\), which would mean that \(s\) did not converge to \(m'\), a contradiction.
