description/proof of that for infinite sequence on partially-ordered set and element of set, if there is any large index whose value is equal to or smaller than element, limit inferior is equal to or smaller than element
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of limit inferior of sequence on partially-ordered set.
Target Context
- The reader will have a description and a proof of the proposition that for any infinite sequence on any partially-ordered set and any element of the set, if the limit inferior exists and there is any large index whose value is equal to or smaller than the element, the limit inferior is equal to or smaller than the element.
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 \(\vert J \vert = \infty\)
\(S\): \(\in \{\text{ the partially-ordered sets }\}\), with any partial ordering, \(\lt\)
\(s\): \(\in \{\text{ the sequences }\}\), such that \(Dom (s) = J\) and \(Ran (s) \subseteq S\)
\(s'\): \(\in S\)
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Statements:
\(\exists lim inf s \land \forall j \in J (\exists j' \in J \text{ such that } j \lt j' (s (j) \le s'))\)
\(\implies\)
\(lim inf s \le s'\)
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2: Note
If \(J\) is finite, it is just a matter of \(s (J_{\vert J \vert}) \le s'\) if and only if \(lim inf s \le s'\), because \(lim inf s = s (J_{\vert J \vert})\).
3: Proof
Whole Strategy: Step 1: see that \(lim inf s \le s'\).
Step 1:
\(lim inf s = Sup (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\).
For each \(m \in \mathbb{N} \setminus \{0\}\), there is an \(n \in \mathbb{N} \setminus \{0\}\) such that \(m \lt n\) and \(s (J_n) \le s'\), by the supposition.
So, for each \(m \in \mathbb{N} \setminus \{0\}\), \(Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \le s (J_n) \le s'\).
So, \(s' \in Ub (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\).
So, \(lim inf s = Sup (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\}) = Min (Ub (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\}))) \le s'\).
