 Sine fixed-point iteration

The sine fixed-point iteration converges to zero from every real starting value. After one step, $|x_n|\le 1$, and $|x_{n+1}|=\sin |x_n|$. Since $0<\sin t<t$ for $0<t\le 1$, the absolute values form a bounded monotone sequence. Its limit $a$ satisfies $a=\sin a$ by continuity, forcing $a=0$. This proves convergence even though the sine is not a contraction mapping on any interval containing zero in its interior.