definition of subsequence of sequence
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of sequence.
Target Context
- The reader will have a definition of subsequence of 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:
\( s\): \(\in \{\text{ the sequences }\}\), with \(Dom (s) = J\)
\( J^`\): \(\subseteq \mathbb{N}\)
\( f\): \(: J^` \to J\), which satisfies the conditions specified below
\(*s^`\): \(= s \circ f\), \(\in \{\text{ the sequences }\}\), with \(Dom (s^`) = J^`\)
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Conditions:
\(\forall j^`_1, j^`_2 \in J^` \text{ such that } j^`_1 \lt j^`_2 (f (j^`_1) \lt f (j^`_2)) \land \forall j \in J (\exists j^` \in J^` (j \le f (j^`)))\)
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2: Note
The condition, \(\forall j \in J (\exists j^` \in J^` (j \le f (j^`)))\), is for not calling \((s_2, s_2)\) for \((s_1, s_2, ...)\) "subsequence", for example: if \((s_2, s_2)\) was called "subsequence", any sequence would have a convergent subsequence just by truncating the sequence finitely.
According to this definition, for \((s_1, s_2)\), \((s_2)\) is a subsequence but \((s_1)\) is not.
