*Cross-posted [from Math SE](https://math.stackexchange.com/questions/3298993/generalized-limits).*

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](https://en.wikipedia.org/wiki/Almost_convergent_sequence) is relevant. However, I don't have the positivity property of [Banach limits](https://en.wikipedia.org/wiki/Banach_limit) since $\operatorname{Lim}(n \mapsto a^n) = (1-a)^{-1}$ which is negative for $a > 1$.