Rigorous proof needed: Acemoglu (Intro Growth) Corollary $2.1.2$ I am reading Acemoglu's intro to modern economic growth. But I am having trouble understanding his proof to a theorem related with stability. Here is the theorem: 


And here are some related definitions and theorems:



My question lies in his proof of corollary $2.1$. How did he show that the sequence $x(t)$ is monotone and bounded above by $x^*$? I think it is not enough to use that the derivative is less than $1$ and the distance to $x^*$ is shrinking to say $x(t)$ is monotone. Can someone give me some more details to understand the proof?

Thanks!
The OP correctly identified a mistake here. Since the author claims monotonicity for a general function, let's disprove it for the simple linear case.
Consider

$$x_{t+1} = g(x_t) = -0.5x_t$$

This satisfies all the requirements and conditions stated by the author, and also that the derivative is smaller than unity in absolute values $\forall x$. But it converges with "damped oscillations", and so it is not monotonic since, say, 

$$x_0 = -3 &lt; 0 = x^*$$
$$x_1 = 1.5$$
$$x_2 = -0.75$$
etc