User blog comment:P進大好きbot/Criterion of the Order-Preserving Property of FGH/@comment-11227630-20191102144819/@comment-35470197-20191102152317

I thought that the most accepted definition of the notion of an FS for a limit term \(x\) requires \(n \mapsto x[n]\) to be strictly increasing.

You do not require it, right? If you do not, then FGH is more likely to break, i.e. can be non-order-preserving. Therefore ar least if we want to make FGH order-preserving, then it is reasonable to require it.