Convergence of $a_{n+1}=x+\sin a_{n}$ Assume $x \in \Bbb R$ is a real number, consider the following sequence:
$a_{n+1}=x+\sin a_{n}, a_0=0.$
For which $x$, the sequence above is convergent $?$ If $x \in \pi \Bbb Z$ then it's obvious, but I wonder what will happen for other $x$.

Suppose that $f(t)=x+\sin(t)$ and $a=f(a)$. According to fixed point theorems, your sequence converges if
$$|f'(a)|<1$$
Which trivially holds if
$$a\notin\pi\Bbb Z$$
But if $a\in\pi\Bbb Z$, well, as you say, the convergence is obvious.
Thus, it converges for any $x\in\Bbb R$.
