How to show $σ^{r}(i) = i$? ( $σ^r$ denotes the power of a relation obtained from repeated composition)

Question

$N$_≤n denotes the set of first n natural numbers i.e., $N$_≤n$ = ${k | k ∈ N ∧ k ≤ n}$. $
An element $a ∈ A$ is called a fixed point of $f$ if $f(a) = a$ where $f: A → A$ is any function over an arbitrary set A. Let σ: N_≤n → N_≤n denote an arbitrary permutation of the set N_≤n.
Show that for all $ i ∈ $N_≤n, there exists a smallest r ∈ N, where 1 ≤ r ≤ n, such that σ^r(i) = i. Note that σ^r denotes the power of a relation obtained from repeated composition.

Answer 1

Note that it suffices to show that there is some positive integer $r$ such that $\sigma^r=\sigma$: if there is one at all, there is certainly a smallest one.

HINT: There are $n!$ permutations of $\Bbb N_{\le n}$. Consider the $n!+1$ permutations $\sigma^r$ for $r=1,\ldots,n!+1$, apply the pigeonhole principle, and prove and use the fact that $\sigma^{r+s}=\sigma^r\circ\sigma^s$ for any positive integers $r$ and $s$.