Difference between revisions of "Subsequence/Definition"
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(Created page with "<noinclude> ==Definition== </noinclude>Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''{{rAPIKM}} as a sequence: * {{M|k:...") |
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Revision as of 15:31, 1 December 2015
Definition
Given a sequence [ilmath](x_n)_{n=1}^\infty[/ilmath] we define a subsequence of [ilmath](x_n)^\infty_{n=1}[/ilmath][1] as a sequence:
- [ilmath]k:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] which operates on an [ilmath]n\in\mathbb{N} [/ilmath] with [ilmath]n\mapsto k_n:=k(n)[/ilmath] where:
- [ilmath]k_n[/ilmath] is increasing, that means [ilmath]k_n\le k_{n+1} [/ilmath]
We denote this:
- [ilmath](x_{k_n})_{n=1}^\infty[/ilmath]
