Can infinitessimals be represented as monotone decreasing sequences?

[jem05]

Hello,
Happy holidays everyone,

I'm trying to prove that any infinitessimal can be written as a monotone decreasing sequence; that is, one of its representations as a sequence of real numbers is a mon. dec. seq.
I'm really stuck, and i don't even know if it's true.
Intuitively, it should work.

I mean i can get a subsequence that is monotone decreasing since the infinitessimal is smaller than any real number,
but how do i know this set of n \in N corresponding to the subsequence chosen \in ultrafilter F.

Any ideas?
Thanks.

 

[fresh_42]

I think if it is not, then it can be used for a refinement of the filter, but there is none by definition of ultrafilter.