= Sine fixed-point iteration
{title2=$x_{n+1}=\sin x_n$}
The <sine fixed-point iteration> converges to zero from every real starting value. After one step, $|x_n|\leq1$, and $|x_{n+1}|=\sin|x_n|$. Since $0<\sin t<t$ for $0<t\leq1$, 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.
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