Let $(F,+_F,*_F, \le_F)$ be a totally ordered field with zero $0_F$ and unity $1_F$. Let $(F,+_F,*_F, \le_F)$ have the least upper bound property.
I know that there is a proof saying that if the field is non-Archimedean, then it doesn't have this property.
I have been wondering if there is another proof of the theorem in title, which, rather then starting with the assumption that a field isn't Archimedean, starts with the assumption that it has L.U.B. property and from that proves that the field is Archimedean.
Does such a proof exist?