description/proof of that for sequence on partially-ordered set and subsequence, if limit superior of sequence exists, limit superior of subsequence does not necessarily exist, but if it exists, it is equal to or smaller than limit superior of sequence
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 subsequence of sequence.
- The reader admits the proposition that for any partially-ordered set, any subset, and any subset of the subset, if the infimum of the subset and the infimum of the subset of the subset exist, the infimum of the subset is equal to or smaller than the infimum of the subset of the subset, and if the supremum of the subset and the supremum of the subset of the subset exist, the supremum of the subset is equal to or larger than the supremum of the subset of the subset.
- The reader admits the proposition that for any partially-ordered set and any \(2\) subsets with any same index set which have some infimums, if for each index, the element of the 1st subset is equal to or smaller than the element of the 2nd subset, the infimum of the 1st subset is equal to or smaller than the infimum of the 2nd subset.
Target Context
- The reader will have a description and a proof of the proposition that for any sequence on any partially-ordered set and any subsequence, if the limit superior of the sequence exists, the limit superior of the subsequence does not necessarily exist, but if it exists, it is equal to or smaller than the limit superior of the sequence.
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 \(Ran (s) \subseteq S\)
\(J^`\): \(\subseteq \mathbb{N}\), such that \(J^` \neq \emptyset\)
\(s^`\): \(= s \circ f\), \(\in \{\text{ the subsequences of } s \text{ with } f: J^` \to J\}\)
//
Statements:
(
\(\exists lim sup s\)
\(\lnot \implies\)
\(\exists lim sup s^`\)
)
\(\land\)
(
\(\exists lim sup s \land \exists lim sup s^`\)
\(\implies\)
\(lim sup s^` \le lim sup s\)
)
//
2: Note
Compare with the proposition that for any convergent sequence on any metric space, its any subsequence converges to the convergence of the sequence.
\(lim sup s = lim sup s^`\) does not necessarily hold.
3: Proof
Whole Strategy: Step 1: see an example that \(lim sup s\) exists but \(lim sup s^`\) does not exist; Step 2: suppose that \(lim sup s\) and \(lim sup s^`\) exist; Step 3: apply the proposition that for any partially-ordered set, any subset, and any subset of the subset, if the infimum of the subset and the infimum of the subset of the subset exist, the infimum of the subset is equal to or smaller than the infimum of the subset of the subset, and if the supremum of the subset and the supremum of the subset of the subset exist, the supremum of the subset is equal to or larger than the supremum of the subset of the subset and the proposition that for any partially-ordered set and any \(2\) subsets with any same index set which have some infimums, if for each index, the element of the 1st subset is equal to or smaller than the element of the 2nd subset, the infimum of the 1st subset is equal to or smaller than the infimum of the 2nd subset; Step 4: see an example that \(lim sup s = lim sup s^`\) does not hold.
Step 1:
Let us see an example that \(lim sup s\) exists but \(lim sup s^`\) does not exist.
Let \(b_0.b_1 b_2 ...\) be the decimal expression of \(\sqrt{2}\).
Let \(J = \mathbb{N}\), \(S = \mathbb{Q}\), \(s: J \to S, j \mapsto b_0.b_1 b_2 ... b_{j / 2} \text{ when } j \text{ is even }; \mapsto 2 \text{ when } j \text{ is odd }\), \(J^` = \mathbb{N}\), and \(s^` = s \circ f\) where \(f: J^` \to J, j^` \mapsto 2 j^`\).
\(lim sup s = Inf (\{Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\})\), \(Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) = 2\) for each \(m \in \mathbb{N} \setminus \{0\}\), and \(Inf (\{Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\}) = 2\).
But \(Sup (\{s^` (J^`_{n^`}) \vert n^` \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n^`\})\) does not exist for each \(m \in \mathbb{N} \setminus \{0\}\), because \(s^`\) approaches \(\sqrt{2}\) from below and does not have any supremum in \(\mathbb{Q}\) (would have the supremum in \(\mathbb{R}\)), 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.
Step 2:
Let us suppose that \(lim sup s\) and \(lim sup s^`\) exist.
Step 3:
For each \(m \in \mathbb{N} \setminus \{0\}\), \(\{s^` (J^`_{n^`}) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le {n^`}\} \subseteq \{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}\), because \(s^` (J^`_{n^`}) = s \circ f (J^`_{n^`})\) but \(f (J^`_{n^`}) = J_n\) where \(n^` \le n\) (refer to Note for the definition of subsequence of sequence), so, \(m \le n\), so, \(s^` (J^`_{n^`}) = s \circ f (J^`_{n^`}) = s (J_n) \in \{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}\).
\(Sup (\{s^` (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \le Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\})\), by the proposition that for any partially-ordered set, any subset, and any subset of the subset, if the infimum of the subset and the infimum of the subset of the subset exist, the infimum of the subset is equal to or smaller than the infimum of the subset of the subset, and if the supremum of the subset and the supremum of the subset of the subset exist, the supremum of the subset is equal to or larger than the supremum of the subset of the subset.
Then, \(lim sup s^` = Inf (\{Sup (\{s^` (J^`_{n^`}) \vert n^` \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n^`\}) \vert m \in \mathbb{N} \setminus \{0\}\}) \le Inf (\{Sup (\{s (J_n) \vert n \in \mathbb{N} \setminus \{0\} \text{ such that } m \le n\}) \vert m \in \mathbb{N} \setminus \{0\}\}) = lim sup s\), by the proposition that for any partially-ordered set and any \(2\) subsets with any same index set which have some infimums, if for each index, the element of the 1st subset is equal to or smaller than the element of the 2nd subset, the infimum of the 1st subset is equal to or smaller than the infimum of the 2nd subset.
Step 4:
Let us see an example that \(lim sup s = lim sup s^`\) does not hold.
Let \(J = \mathbb{N}\), \(S = \mathbb{R}\), \(s: J \to S, j \mapsto - 1 \text{ when } j \text{ is even }; \mapsto \text{ 1 } \text{ when } j \text{ is odd }\), \(J^` = \mathbb{N}\), and \(s^` = s \circ f\) where \(f: J^` \to J, j^` \mapsto 2 j^`\).
Then, \(lim sup s = 1\), but \(lim sup s^` = - 1\), and \(lim sup s^` \lt lim sup s\).
