Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers

Abstract

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers $\widetilde{\mathbb{R}}$ does not generalize classical results. E.g.~the sequence $\frac{1}{n}\not\to0$ and a sequence $(x_{n})_{n\in\mathbb{N}}$ converges \emph{if} and only if $x_{n+1}-x_{n}\to0$. This has several deep consequences, e.g.~in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that $\widetilde{\mathbb{R}}$ is necessarily not a complete ordered set, e.g.~the set of all the infinitesimals does not have neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.