convergence of continued nested function Does $$\sin (x + \sin (x + \sin (x + \sin (x + ... ) ) )))  $$ converge to any limit? If so, to what?
Recursive plotting appears at times to come to same/similar profiles.

Define the sequence
$$x_{n+1} = x_0 + \sin x_n$$
For any $x_0$, the function $x_0+\sin x$ is $1$-lipschitzian and maps the interval $[x_0 - 1, x_0 + 1]$ into itself. Therefore it has a unique fixed point in that interval, and the sequence $x_n$ converges to it according to this modification of the Banach fixed point theorem.
