Can infinitessimals be represented as monotone decreasing sequences?

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This discussion centers on the representation of infinitesimals as monotone decreasing sequences within the context of real numbers. The user seeks to prove that any infinitesimal can indeed be expressed as such a sequence, acknowledging the challenge of establishing a corresponding set of natural numbers within an ultrafilter. The conversation highlights the relationship between infinitesimals and ultrafilters, suggesting that if a monotone decreasing sequence representation is not possible, it could lead to a refinement of the ultrafilter, which contradicts its definition.

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Mathematicians, students of advanced calculus, and researchers in set theory and non-standard analysis will benefit from this discussion, particularly those interested in the properties of infinitesimals and their representations.

jem05
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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.
 
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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.
 

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