I'm not saying you shouldn't find a counterexample. What I am saying is that your counterexample may need to be touched up in rigor a bit, depending on the standards of your teacher. Rigorous inequalities can help you in this regard.

As for the derivative approach, if the derivative exists everywhere and is less than 1 in magnitude, then the function is Lipschitz, of Lipschitz constant less than 1. Then the

Banach fixed point theorem applies. That is, having a derivative everywhere that's less than 1 in magnitude is a stronger condition than being Lipschitz.

Overall, what you're trying to do is show that the Banach fixed point theorem does not apply to this function, because this function doesn't satisfy the assumptions of the theorem.