4 hypothesis about iterated functions

[Shanghai46]

ONE : let's assume there's a function \(f\) that has an attractive fixed point \(\tau\). Let's consider the real interval \(I\) which is the biggest monotomic interval so that \(\forall x_0\in I\), the infinite iteration of \(f(x_0)\) converges towards \(\tau\), and that all positive iterations of \(f(x_0)\) are inside the interval \(I\).
is \(\forall n\in\mathbb{N}, |f^{\circ n+1}(x_0)-\tau| < |f^{\circ n}(x_0)-\tau|\) true? (kinda answered by Tommy but without evidence..... So......)

Also wrong, I found a counter example