How to prove the periodicity of an iterated function?
For example, how to prove $\sin_{[n]}(x)$ is a periodic function of period $2\pi$ $\forall n\in\mathbb{R}^+\cap\{0\}$ ?
How to prove the periodicity of an iterated function?
For example, how to prove $\sin_{[n]}(x)$ is a periodic function of period $2\pi$ $\forall n\in\mathbb{R}^+\cap\{0\}$ ?
In total generality, if $f: \mathbb{R} \rightarrow \mathbb{R}$ is periodic with period $p$, i.e. $f(x+p) = f(x)$ for all $x$, then on the nose $g \circ f$ has period $p$ for any $g$.