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### prove that the sequence given by recurrence converges [duplicate]

Given a: $a_0>0$, and $a_{n+1}=\sin(a_n)$. I need to prove that the sequence $a_{n=0}^\infty$ converges, then calculate the limit. I have proved that is crescent, then I got stuck.
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### Repeated applications of cos function converges around $0.7390851$ [duplicate]

I set my calculator to radian mode. I enter any random number $x$. Then I repeatedly apply cosine to that number, getting the sequence $$\cos x, \cos(\cos x), \cos(\cos(\cos x)) \ldots$$ and so on. No ...
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### what are the properties of this sequence? [closed]

Can we assert that this sequence is convergent or divergent? $$u_o = a, \text{ and } \forall n \in \mathbb{N} : u_{n+1} = \sin(u_n)$$ What is the limit of $u_{n}$ if a=$\frac{\pi}{4}$