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