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# Sequence behavior watch

1. Is it possible to help me expressing this in the right way? I know what's happening but I am struggling to express it as correct as possible. The theorem doesn't state that if a sequence is monotonically increasing or decreasing and bounded above or below then the sequence converges to a finite number either diverges, but from part a) the sequence must converge to 1. Can you express it in your way so I can understand? Thanks again for another one time!!
2. It's literally just:

x_n is monotonic increasing and bounded above, therefore converges to a limit L. By the first part of the question, L = 0 or L = 1. Since x_n >= x_0 > 0, we must have L > 0 and so L = 1.
3. Thank you very much for the help. Completely understand it! Appreciate it!
4. Sorry again, just a last small question! When x(0)>2 I know that the first iteration will produce an 0<x(1)<1 and after that we will have a monotonically increasing sequence bounded above by 1 and by MST the sequence converges to a finite limit L which is 1 since x(0)>2 => L>0. Is this the correct way to write it?
5. Why does x(0) > 2 imply L > 0?
6. Because we will have a monotonically increasing function after the first iteration so how can L be 0?
7. But then what you're really saying is: "if x(0) > 2 then 0 < x(1) < 1 and by the previous result this implies x(n) -> 1". (In other words, the key point is that x(1) < 1, NOT that x(0) > 2. Writing "x(0) > 2 => L > 0" is very liable to misinterpretation here).
8. Correct, thanks again! Appreciate it!!

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Updated: March 25, 2011
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