How do you distinguish a different proof from the same proof in different language? My point is that the concepts simplify the language of the proof and put it in an intuitive context.

@Zen: Are there jobs where a degree in Mathematics is a disadvantage? I can certainly see where it could be neutral. Do you have an example in mind where it actually hurts?

From Mark's answer I see that you mean I am missing a direction. I wasn't aware that the direction was oriented in this manner; I assumed it meant unoriented direction, i.e., slope. In this case, I am even more interested to know the answer to both questions.

Which interval is that? This should hit everything. It might be good to clarify the question since two people have answered the original question. I assumed you were looking for any decomposition, including using closed or open intervals.

You are correct. I'm still figuring all this out, but I'm working on a project where it would be nice to have specific examples. Thanks!