If $x = \sin(x)$, then is it true that $x = \sin(\sin(\sin(\sin(\cdots(\sin(x))\cdots))))$?

Question

If $x = \sin(x)$, then is the following true? $$x = \sin(\sin(\sin(\sin(\cdots(\sin(x))\cdots))))$$

I don't see a particular reason for not believing it.

Answer 1

More generally, if $a=f(a)$, then $a=f(f(f(\cdots f(a)\cdots)))$. This is easily proved by induction: $$ f^{n+1}(a)=f(f^n(a))=f(a)=a $$ This holds for arbitrary sets $A$ and functions $f: A \to A$.