Cross-posted from Math SE.

The linked question explores the concept of a "generalized limit" that assigns values to sequences which diverge in the Cauchy sense. One question I asked is whether there exists a generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ such that

1. $\operatorname{Lim}$ is stronger than the stable and linear closure of the Cauchy limit.

2. $\operatorname{Lim}$ can be explicitly defined in terms of the Cauchy limit, e.g. $\operatorname{Lim} = \lim \circ f$ where $f : X^\mathbb{N} \rightarrow X^\mathbb{N}$ is some sequence transform.

I'd be curious to know if anyone knows what such a $\operatorname{Lim}$ might be (if it exists). Even an explicit characterization of a generalized limit satisfying 1 alone would be interesting.