Rigorous proof needed: Acemoglu (Intro Growth) Corollary $2.1.2$
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