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 \(\frac{1}{n}\not \rightarrow 0\) and a sequence \((x_{n})_{n\in \mathbb {N}}\) converges if and only if \(x_{n+1}-x_{n}\rightarrow 0\). 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 has 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.

众所周知，Colombeau 广义数序列\(\widetilde{\mathbb {R}}\)的尖锐拓扑中的极限概念不能推广经典结果。例如序列\(\frac{1}{n}\not \rightarrow 0\)和序列\((x_{n})_{n\in \mathbb {N}}\)收敛当且仅当\ (x_{n+1}-x_{n}\rightarrow 0\)。这有几个深刻的后果，例如在研究级数、解析广义函数，或广义函数积分中的 sigma 可加性和经典极限定理。缺乏这些结果也与\(\widetilde{\mathbb {R}}\)不一定是完整的有序集，例如所有无穷小的集合既没有上界也没有下界。我们通过引入超自然数、超序列、关闭上层和下层的概念来提出这些问题的解决方案。通过这种方式，我们可以推广超序列的超极限的所有经典定理。该论文探讨了可以应用于其他非阿基米德设置的想法。