Ordering a sequence

  • Thread starter daniel_i_l
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  • #1
daniel_i_l
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Main Question or Discussion Point

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]?
 

Answers and Replies

  • #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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