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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: enter image description here

And here are some related definitions and theorems: enter image description here enter image description here

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!

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  • $\begingroup$ 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. $\endgroup$ – VCG Sep 16 '16 at 1:49
  • $\begingroup$ 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 $x^*$ $\endgroup$ – ask Sep 16 '16 at 3:07
  • $\begingroup$ And also we are in the non-linear case... $\endgroup$ – ask Sep 16 '16 at 3:16
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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,

$$x_0 = -3 < 0 = x^*$$ $$x_1 = 1.5$$ $$x_2 = -0.75$$ etc

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  • $\begingroup$ I believe you meant damped oscillations (reduced over time) rather than dumped $\endgroup$ – Three Diag Oct 10 '16 at 10:23
  • $\begingroup$ @ThreeDiag Thanks for spotting this. Fixed. $\endgroup$ – Alecos Papadopoulos Oct 10 '16 at 23:14

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