Ordering a Sequence: Can We Always Change the Order?

[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)}?

 

[HallsofIvy]

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! {a_n}= 1/n, for example, has no smallest member and so cannot be "reordered" to be increasing.