# Ordering a sequence

1. May 5, 2008

### daniel_i_l

Lets say we have a sequence of reals. Is it always possible to change the order to that for all n $$a_{n+1} >= a_n$$?
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
$$a_{f(n+1)} >= a_{f(n)}$$?

2. May 5, 2008

### HallsofIvy

Staff Emeritus
No. That would imply that $a_n\le a_m$ 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.