Question about the properties of iterated functions

[Shanghai46]

Let's take the function \(f\), which has a fixed point \(\tau\). Let's also consider a real number \(x_0\) that belongs to the biggest monotonic interval of \(f\) that contains \(\tau\) such that the infinite iteration of \(f(x_0)=\tau\), and that for all \(x_0\) that belongs to that interval, \(f(x_0)\) also belongs to that interval.

In this case, I just wonder if the distance between the \(n\)th iteration of \(f(x_0)\) and \(\tau\) keeps decreasing as \(n\) increases. \(\forall n\in\mathbb{N}, |f^{n+1}(x_0)-\tau|<|f^{n}(x_0)-\tau|\).

Is it true for all functions and starting number with these restrictions, or do we need other restrictions to make it always true?

UPDATE : if \(f'(\tau)<0\), it is false since I managed to find a counter example
                 if \(f'(\tau)>0\), it is true, I managed to demonstrate it!