Let $A$ be a finite set. Let $f:A\to A$ be an injective function from the set $A$ to itself and let $x_0$ be an element of $A$. Let $x_n$ be defined for all positive integers $n$ so that $x_n=f(x_{n-1})$ for all positive integers $n$. Prove that there exists some positive integer $m$ such that $x_m=x_0$.
Now a function is said to be injective when there is a one to one mapping. When $f : A \to A$, it means that for any element $x$ in $A$, it maps to a distinct element $y$ in $A$.
Now applying the function $x_n=f(x_{n-1})$ if $n =1$, then the function would give a result $x_0$.
The question is how do i prove this even thou i have show that there exists some positive integer which in this case is $1$