  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.
