[John at 21](https://forum.azimuthproject.org/discussion/comment/16715/#Comment_16715), you write

> 2) However, g(b) is in the set {a∈A:f(a)≤Bb}, i.e. f(g(b))≤Bb

> Why? Because

> f(g(b))≤Bb if and only if g(b)≤Ag(b).
> So, any lower bound of this set must be ≥g(b). Thus, g(b) is a least upper bound.

Shouldn't that be *"So, any **upper** bound of this set ..."*? I may have gotten thoroughly confused about up and down, but I thought you were talking about all the other upper bounds having to be larger than \\(g(b)\\).