Question: How do I even begin to do this problem? I don't know where to even begin. 

The professor of the class tried to give us hints (as this is a redo to our homework) and said "The contrapositive is 'If f is not one to one, then f is not increasing.' Write out carefully and clearly what it means to say that f is not one to one and that f is not increasing. Then using the hypothesis, prove f is not increasing." 

I am even more lost now that he has given us this hint, any idea on how to start this proof?

*Edit - I'm not asking for you guys to do the proof, I just don't know where to start and my professor doesn't answer any of my emails, so I'm turning to you guys.*

Answer: 
By definition, a (strictly) increasing function is such that $x<y\implies f(x)<f(y)$. In particular, if $x\neq y$; either $x<y$ or $y<x$. In any case, what do you conclude about $f(x)$ and $f(y)$?
Assume that $f$ is not one-to-one. Then there exist $x, y \in I$ such that $x \neq y$ and $f(x) = f(y)$..