Sequential convergence in non-standard analysis

Question

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.

Answer 1

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.

Answer 2

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.