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. It asks the following question:

Does there exist an "explicitly definable" generalized limit $\operatorname{Lim} : X^\mathbb{N} \rightharpoonup X$ that is stronger than the linear and stable closure of the Cauchy limit?

I'd be curious to know what such a $\operatorname{Lim}$ might be.

As Gerald pointed out, the concept of an almost convergent sequence is relevant. However, I don't have the positivity property of Banach limits since $\operatorname{Lim}(n \mapsto a^n) = (1-a)^{-1}$ which is negative for $a > 1$.