Ah. The conditions in your theorems ARE Lipschitz conditions of a particular kind. Look

here for a Lipschitz condition definition. You can see that the conditions in your theorems are that the function is Lipschitz with Lipschitz constant in the interval (0,1). If you look at the

first property in the Properties section of the wiki, you will see that an everywhere differentiable function is Lipschitz if and only if its derivative is bounded, and it tells you what the Lipschitz constant is in that case: the least upper bound of the magnitude of the derivative! So there's a direct tie-in to the conditions of your theorems. Does that help?

So your logic would work like this:

1. The function is everywhere differentiable.

2. The magnitude of the derivative is bounded by 1 - epsilon everywhere on the interval, for some epsilon greater than zero.

3. Therefore, the function is Lipschitz with a Lipschitz constant in the interval (0,1).

4. Therefore, the conditions of the theorem you've been given apply.

5. Therefore, the conclusion of the theorem holds.

And you're done. The major step here is # 2. What's epsilon?