Here's some ideas on how to prove them:

1. If for some , then f'(x) would have to be greater than r for all , which would mean it is unbounded as .

2. If the sup is not zero, then it must be less than zero. So the derivative is less than some fixed negative number, which again means f(x) is unbounded.

3. This follows immediately from #1 and #2.

- Hollywood