Studying convergence of recursively defined sequence $a_1=2\text{,}\; a_{n+1}=2\sin(a_n)$ Firstly sorry for duplicate if this was asked before, i couldnt find.
This sequence is not monotone, but it seems convergent, i have plotted with Maple. Any hints to prove this is a Cauchy sequence? Or another method?
By the way, sequence seems divergent if i increase the factor 2. For example $a_1=2\text{,}\; a_{n+1}=2.25\cdot\sin(a_n)$ is divergent.
Thanks..

Hints:


*

*Show that $|a_n|\le2$ for all $n$

*Show that if $a_n\to L, L=2\sin L$

*Determine the stability of the fixed points of the map $x\to 2\sin x$

*Deduce the convergence behaviour of $a_n$

