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Ordering a sequence

  1. May 5, 2008 #1

    daniel_i_l

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    Gold Member

    Lets say we have a sequence of reals. Is it always possible to change the order to that for all n [tex]a_{n+1} >= a_n[/tex]?
    Or in other words,
    Does there always exist a bijective function:
    f:Z->Z (where Z is the set of positive natural numbers) so that for all n
    [tex]a_{f(n+1)} >= a_{f(n)}[/tex]?
     
  2. jcsd
  3. May 5, 2008 #2

    HallsofIvy

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    No. That would imply that [itex]a_n\le a_m[/itex] as long as m> n. In particular, a1 would have to be the smallest number in the list- and the list may not have a smallest member! {an}= 1/n, for example, has no smallest member and so cannot be "reordered" to be increasing.
     
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