Question 86
Let $\mathbb{Z}_{ \gt 0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{ \gt 0} \rightarrow \mathbb{Z}_{ \gt 0}$. For any $m, n \in \mathbb{Z}_{ \gt 0}$ we write $f^{n}(m)=\underbrace{f(f(\ldots f}_{n}(m) \ldots))$. Suppose that $f$ has the following two properties:
(i) If $m, n \in \mathbb{Z}_{ \gt 0}$, then $\frac{f^{n}(m)-m}{n} \in \mathbb{Z}_{ \gt 0}$;
(ii) The set $\mathbb{Z}_{ \gt 0} \backslash\left\{f(n) \mid n \in \mathbb{Z}_{ \gt 0}\right\}$ is finite.
Prove that the sequence $f(1)-1, f(2)-2, f(3)-3, \ldots$ is periodic.
(i) If $m, n \in \mathbb{Z}_{ \gt 0}$, then $\frac{f^{n}(m)-m}{n} \in \mathbb{Z}_{ \gt 0}$;
(ii) The set $\mathbb{Z}_{ \gt 0} \backslash\left\{f(n) \mid n \in \mathbb{Z}_{ \gt 0}\right\}$ is finite.
Prove that the sequence $f(1)-1, f(2)-2, f(3)-3, \ldots$ is periodic.