Are all Hyperreal Infinitesimals representable by Monotonically Decreasing Sequences to 0? I know there are many possible theoretical ways to built *R, including axiomatic and set-theoretic approaches.  I am limiting my attention specifically to the Superstructure approach, perhaps best outlined by Robert Goldblatt's "Lecture on the Hyperreals": involving infinite sequences of real numbers, using a non-principal ultrafilter U to combine them into representative classes.  Thus any hyperreal h can be represented by a sequence $\langle h_0, h_1, h_2, \ldots \rangle$.  (And, of course, this representation is not unique, as there are an infinite number of such representations, for each h.)
Since all limited hyperreals can be uniquely described as r + e (r a real number, e an infinitesimal), my interest is on infinitesimals.  Specifically, my question is:

*

*Is it true that every positive infinitesimal can be represented by a monotonically decreasing sequence converging to 0?  Or more formally: Given any positive infinitesimal e, does there exist a representation $\langle e_0, e_1, e_2, \ldots \rangle$ of e, such that: ei > ei+1 for all i in N, en's all positive, and lim en = 0?

Certainly some of them can (such as: $\langle 1, 1/2, 1/3, \ldots \rangle$).  But are we guaranteed that this is true in all cases?  And I realize that the choice of Ultrafilter U may play a part.  It seems to me the answer could be one of the following:

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*Yes, this is always true.

*No, this is always false.

*No, it is false, but could be made true if we dropped the requirement of monotonicity.

*It entirely depends on the choice of U.

*It is unknown if this is true or false (or if the choice of U plays a role).

(In the event of #4 as the answer, is there anything we can say about this Ultrafilter choice?)
I am not making any assumptions about the Continuum Hypothesis, Martin's Axiom, or any other axiom at this time.  (Although I would assume that CH would have to be false in the case of #4, since CH $\to$ *R models are isomorphic.)  This is more for an understanding on my part as to how generalized infinitesimals can be, relative to real convergent sequences to 0.
Thanks.
 A: Great question! The answer is that it can depend on the ultrafilter. (Note that even if $\mathbb{R}^\mathcal{U}\cong\mathbb{R}^\mathcal{V}$ the ultrafilters $\mathcal{U}$ and $\mathcal{V}$ may yield very different representation systems, so in fact $\mathsf{CH}$ does not immediately trivialize things.)
Suppose $\alpha=(a_i)_{i\in\mathbb{N}}$ is a sequence of positive reals with the following properties:

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*$\lim_{i\rightarrow\infty}a_i=0$.


*There is no finite $S_1,...,S_n\subseteq\mathbb{N}$ such that $\mathbb{N}=S_1\cup ...\cup S_n$ and on each $S_i$ the sequence $\alpha$ is monotonic.
Constructing such an $\alpha$ (very explicitly! no complicated set theory needed) is a fun exercise:

 Consider something like $$1,\quad{1\over 3}, {1\over 2}, \quad{1\over 4},\quad {1\over 7},{1\over 6},{1\over 5},\quad {1\over 8},\quad ...,$$ which goes to zero but has arbitrarily large finite increasing subsequences.

The first bulletpoint ensures that such an $\alpha$ names an infinitesimal in every ultrapower of $\mathbb{R}$, but by the second bulletpoint we can whip up an ultrafilter $\mathcal{U}$ such that for each $S\subseteq\mathbb{N}$ if $\alpha$ is monotonic on $S$ then $S\not\in\mathcal{U}$, and this $\mathcal{U}$ will have the property that $\alpha\not\sim_\mathcal{U}\beta$ for any monotonic $\beta$.
On the other hand, every Ramsey ultrafilter will have the property that every infinitesimal has a "monotonic name." Note that the existence of Ramsey ultrafilters follows from $\mathsf{CH}$, so - elaborating on my parenthetical remark above - $\mathsf{ZFC+CH}$ proves both that all ultrapowers of $\mathbb{R}$ over $\mathbb{N}$ are isomorphic and that some but not all ultrafilters on $\mathbb{N}$ have the "all-infinitesimals-are-monotonically-named" property!
Finally, it turns out that the property "all-infinitesimals-have-monotonic-names" is equivalent to P-point-ness, and $\mathsf{ZFC}$ can't prove P-points exist. See Cutland/Kessler/Kopp/Ross 1988, which I learned about from Mikhail Katz's answer to the above-mentioned near-duplicate. (EDIT: Corazza's 2015 paper $P$-points in the construction of the real line seems to be a rediscovery of this CKKR result. I wouldn't be surprised if it's been rediscovered several times since both the problem and its solution are quite natural.)
