> 
> We look at the equation $y'=\sin(y)$, if $y$ is a solution which maintain 
>  $y(0) \in (\pi,2\pi)$, without solving the equation, prove that the limit 
>  $\lim\_{x\to\infty} y(x) $ exist and calculate it.
> 
> 
> 

Is it enough quoting the Existence and Uniqueness Theorem for proving existence and drawing the graph for calculating the limit? (I do not see any other way to find the limit without calculating it)