Any ideas? Thanks!
Let I be a bounded interval. Prove that the intersection of I and Z is a finite set.
Maybe I'm taking some theorems for granted, here, which you are not permitted to do. However, I would say simply, as follows:
Since $\displaystyle I$ is bounded, then so too is $\displaystyle I\cap\mathbb{Z}$ bounded. Thus $\displaystyle |I\cap\mathbb{Z}|=1+\sup(I\cap\mathbb{Z})-\inf (I\cap\mathbb{Z})$.