# 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!

• Given that $|a|<1$ and the linear difference is affine, ratio being less than 1 (dividing the RHS over to the left) is enough to get monotone. Try some examples. – VCG Sep 16 '16 at 1:49
• It only says the distance is decreasing, and it does not eliminate the case say $x(0)=1, x^*=3, x(1)=4, x(2)=2.5, x(3)=3.25...$, or even an extreme case that $x^*=1, x(2k-1)=\frac{1}{2}- \frac{1}{2k-1}, x(2k)=\frac{3}{2}+\frac{1}{2k}$, where $x(t)$ does not converge to $x^*$ – ask Sep 16 '16 at 3:07
• And also we are in the non-linear case... – ask Sep 16 '16 at 3:16

$$x_{t+1} = g(x_t) = -0.5x_t$$
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,
$$x_0 = -3 < 0 = x^*$$ $$x_1 = 1.5$$ $$x_2 = -0.75$$ etc