I have the following problem that I am stuck on:
Let $\mathbb{F}$ be an ordered field with the property that if $[a,b]$ is any closed interval and $f:[a,b]\to\mathbb{F}$ is any continuous function such that $f(a)>0$ and $f(b)<0$, then there is an $x\in (a,b)$ such that $f(x)=0$. Show that such a field $\mathbb{F}$ has the least upper bound property.
My Strategy/Thoughts: Am I correct in thinking that this is just showing that the Intermediate Value Theorem implies the Completeness Axiom? If so, how do I go about it? I was thinking that I could assume that the Intermediate Value Theorem holds for some $f$ on the field, and assume that the Completeness Axiom does not necessarily hold. Then I think I could reach a contradiction by constructing a function $f$ which is not continuous. Would that work?
Thanks in advance for any help and suggestions!
Let $A$ be a nonempty subset of $F$ with an upper bound, but with no least upper bound. Let $B$ be the set of upper bounds of $A$. Investigate $$ f(x) = \begin{cases} -1&x\in B,\\ 1&x\notin B. \end{cases}$$