definition of limit inferior of sequence on partially-ordered set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of infimum of subset of partially-ordered set.
- The reader knows a definition of supremum of subset of partially-ordered set.
Target Context
- The reader will have a definition of limit inferior of sequence on partially-ordered set.
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 \(J \neq \emptyset\)
\( S\): \(\in \{\text{ the partially-ordered sets }\}\) with any partial ordering, \(\lt\)
\( s\): \(\in \{\text{ the sequences }\}\), such that \(Dom (s) = J\) and \(Ran (s) \subseteq S\)
\(*lim inf s\): \(= s (J_n)\) when \(\vert J \vert = n \in \mathbb{N} \setminus \{0\}\); \(= Sup (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\) otherwise
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Conditions:
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2: Note
When \(J\) is infinite, \(lim inf s\) does not necessarily exist, because \(Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\})\) may not exist for an \(m\) and \(Sup (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\) may not exist even if all the \(Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\})\) s exist.
For example, for \(S = \mathbb{Q}\), for the decimal expression, \(- \sqrt{2} = - d_0. d_1 d_2 ...\), \(J = \mathbb{N}\), and \(s (0) = - d_0\), \(s (1) = - d_0. d_1\), \(s (2) = - d_0. d_1 d_2\), ..., does not have any \(Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\})\) in \(\mathbb{Q}\) for each \(m\).
