Let $f$ be a one-to-one function from $X=\{1,2,\dots,n\}$ onto $X$. Let $f^k=f\circ f\circ \cdots \circ f$ denote the $k$-fold composition of $f$ with itself.
- Show that there are distinct positive integers $i$ and $j$ such that $f^i(x)=f^j(x)$ for all $x\in X$.
- Show that for some positive integer $k,~~~f^k(x)=x$ for all $x\in X$.