Lemma: Let be a transitive relation on the interval . If each has a neighborhood such that whenever and , then .

First, I am not sure how exactly to prove the lemma, which is the first part of my problem. Here are my thoughts:

Let $\displaystyle x\in[a,b]$ be arbitray and $\displaystyle \rho$ be a transitive relation on $\displaystyle [a,b]$. By our hypothesis, we can find a neighborhood of $\displaystyle x$ such that whenever and . But, if we condsider the set $\displaystyle A=\{x\in[a,b]|a\rho\\x\}$ it is clear that $\displaystyle sup(A)$ exists and because $\displaystyle [a,b]$ is compact $\displaystyle b=sup(A)$. Since $\displaystyle u$ must share a transitive relation with $\displaystyle a$ and $\displaystyle v$ a transitive relation with $\displaystyle b$, it must be that case that $\displaystyle a\rho\\b$.

IVT proof: The thing is, I know how to prove the IVT without the use of this lemma, but I end up getting all mixed when I try to apply it. I know that I have to find the transitive relation that works, so I have been trying to use $\displaystyle <$ alone, but I need to add on some more conitions to the relation. So, basically I need some help finding the relation and how to apply it.

Thanks alot