Question 92
Let $\mathbb N$ be the set of positive integers. Let $f:\mathbb N\to\mathbb N$ satisfy, for every $n\in\mathbb N$,
$$
f(n)-n\lt2021\qquad\text{and}\qquad f^{\circ f(n)}(n)=n,
$$
where $f^{\circ r}$ denotes $r$ successive applications of $f$. Prove that $f(n)=n$ for infinitely many $n\in\mathbb N$.
$$
f(n)-n\lt2021\qquad\text{and}\qquad f^{\circ f(n)}(n)=n,
$$
where $f^{\circ r}$ denotes $r$ successive applications of $f$. Prove that $f(n)=n$ for infinitely many $n\in\mathbb N$.