I'm trying to prove that; If any Cauchy sequence is convergent in an ordered field F, every nonempty subset of F that has an upperbound has a sup in F.
Let A be a nonempty subset of F that is not a singleton and has an upperbound in F. Let $a_0 \notin v(A)$ and $b_0 \in v(A)$. It's written in my book that for every $e \in P_F$, there exists $N \in \omega$ such that $N≧(b_0 - a_0)/e$.
I think this is not accurate since it hasn't showed that such F is Archimedean.. Is such F archimedean? Or in such a condition does there exist such N?
-Definition of a Cauchy sequence; For every $e\in P_F$, there exists $N\in \omega$ such that if $i,j≧N$, then $|x(i) - x(j)| < e$. ($x:\omega \to F$ is a sequence)