Proving the IVP using the lemma...

Lemma: Let http://www.mathhelpforum.com/math-he...e3547a85-1.gif be a transitive relation on the interval http://www.mathhelpforum.com/math-he...edadca7e-1.gif. If each http://www.mathhelpforum.com/math-he...d5261997-1.gif has a neighborhood http://www.mathhelpforum.com/math-he...cd2579eb-1.gif such that http://www.mathhelpforum.com/math-he...909ff1bc-1.gif whenever http://www.mathhelpforum.com/math-he...33feab0f-1.gif and http://www.mathhelpforum.com/math-he...be9e57ea-1.gif, then http://www.mathhelpforum.com/math-he...69c28225-1.gif.

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 http://www.mathhelpforum.com/math-he...909ff1bc-1.gif whenever http://www.mathhelpforum.com/math-he...33feab0f-1.gif and http://www.mathhelpforum.com/math-he...be9e57ea-1.gif. 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