description/proof of that for sequence on real numbers set with canonical ordering, if limit superior exists, limit inferior does not necessarily exist, and if limit inferior exists, limit superior does not necessarily exist
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of limit superior of sequence on partially-ordered set.
- 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 a sequence on the real numbers set with the canonical ordering, if the limit superior of the sequence exists, the limit inferior of the sequence does not necessarily exist, and if the limit inferior of the sequence exists, the limit superior of the sequence does not necessarily exist.
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 \(J \neq \emptyset\)
\(\mathbb{R}\): with the canonical ordering, \(\lt\)
\(s\): \(\in \{\text{ the sequences }\}\), such that \(Dom (s) = J\) and \(Ran (s) \subseteq \mathbb{R}\)
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Statements:
Not necessarily "\(\exists lim sup s \implies \exists lim inf s\)"
\(\land\)
Not necessarily "\(\exists lim inf s \implies \exists lim sup s\)"
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2: Note
When we say "\(lim sup s\) does not exist" or "\(lim inf s\) does not exist", that means that it does not exist in \(\mathbb{R}\): someone may say that it exists as \(\infty\) or \(- \infty\), but not \(\infty\) nor \(- \infty\) is any element of \(\mathbb{R}\): that someone is really thinking of a sequence on the extended real numbers linearly-ordered set, \(\mathbb{R} \cup \{- \infty, \infty\}\), not on \(\mathbb{R}\), even if \(s\) does not take \(\infty\) or \(- \infty\).
For a sequence on a linearly-ordered set, \(s\), the existence of \(lim sup s\) does not imply the existence of \(lim inf s\) and the existence of \(lim inf s\) does not imply the existence of \(lim sup s\), because \(\mathbb{R}\) is a linearly-ordered set.
For a sequence on a partially-ordered set, \(s\), the existence of \(lim sup s\) does not imply the existence of \(lim inf s\) and the existence of \(lim inf s\) does not imply the existence of \(lim sup s\), because \(\mathbb{R}\) is a partially-ordered set.
3: Proof
Whole Strategy: Step 1: see an example that \(lim sup s\) exists but \(lim inf s\) does not exist; Step 2: see an example that \(lim inf s\) exists but \(lim sup s\) does not exist.
Step 1:
Let us see an example that \(lim sup s\) exists but \(lim inf s\) does not exist.
Let \(J = \mathbb{N}\) and \(s: j \mapsto - 1 \text{ when } j \text{ is even }; \mapsto - 2^j \text{ when } j \text{ is odd }\).
Then, \(lim sup s\) exists, because \(Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) = - 1\) for each \(m \in \mathbb{N} \setminus \{0\}\), and \(lim sup s = Inf (\{Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\}) = - 1\).
But \(Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\})\) does not exist in \(\mathbb{R}\) for each \(m \in \mathbb{N} \setminus \{0\}\), so, \(lim inf s = Sup (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\) does not exist.
Step 2:
Let us see an example that \(lim inf s\) exists but \(lim sup s\) does not exist.
Let \(J = \mathbb{N}\) and \(s: j \mapsto 1 \text{ when } j \text{ is even }; \mapsto 2^j \text{ when } j \text{ is odd }\).
Then, \(lim inf s\) exists, because \(Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) = 1\) for each \(m \in \mathbb{N} \setminus \{0\}\), and \(lim inf s = Sup (\{Inf (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\}) = 1\).
But \(Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\})\) does not exist in \(\mathbb{R}\) for each \(m \in \mathbb{N} \setminus \{0\}\), so, \(lim sup s = Inf (\{Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\) does not exist.
