# Sequential convergence in non-standard analysis

Is the following way of defining sequential convergence correct? The real sequence $(x_n)$ converges to a real number $L$ if the hyperreal number $(x_1 - L, x_2-L, x_3-L,\ldots )$ is an infinitesimal.

• .....or if it's zero, depending on whether or not you include 0 among the infinitesimals. – DanielWainfleet May 2 '17 at 16:33

I assume you are working within the ultrafilter approach to nonstandard-analysis. Then this does not work; see for instance:

Let $U$ be the ultrafilter (on $\mathbb{N}$) corresponding to said hyperreal system. Let $A$ be an infinite but not cofinite subset of $\mathbb{N}$ which belongs to $U$. Such a set exists: either $2\mathbb{N}$ or $2\mathbb{N}+1$ is an example.

Let $x: \mathbb{N}^* \rightarrow \mathbb{R}$ be the sequence such that $x(n) = L + \frac{1}{n}$ if $n-1 \in A$ and $x(n) = L+1$ otherwise. Then $[(x_1-L,x_2-L,...)]$ is an infinitesimal despite the fact that $x$ does not converge to $L$.

The reciprocal is true however: if the sequence converges to $L$, the described hyperreal number is infinitesimal.

In Internal Set Theory, the usual notion is near covergence. A sequence $\{x_n\}$ nearly converges to a real number $c$ if $|x_k - c|$ is infinitesimal whenever $k$ is nonstandard.

It is straightforward to show that all standard sequences converge (in the classical sense) if and only if they nearly converge.

• pash, I don't think the OP is working with nonstandard indices. – Mikhail Katz May 4 '17 at 8:54
• What references call this near convergence? – GPhys May 5 '17 at 1:35
• @MikhailKatz, no, but I think it's useful to offer the Nelsonian approach, even if the question takes a Robinsonian perspective. – pash May 9 '17 at 7:03
• @GPhys, most of the standard IST references. The terminology is Nelson's—he consistently uses the modifiers near and nearly for analogues of classical definitions that may differ for nonstandard objecets but which are equivalent for standard objects. If you're looking for an easy introduction to the axiomatic approach, I recommend Alain Robert's Nonstandard Analysis from Dover. – pash May 9 '17 at 7:12
• @pash Can you list some? Alain and the Nelson articles I've read don't use it, but I think Nelson changed his terminology a lot through the years. (I'm asking just to find the references I haven't read, so please share!) – GPhys May 9 '17 at 10:32