Given a set $X$ of finite size and a function $f: X \to X$ such that there exists a positive $k \in \mathbb{Z}$ such that $|f^k(X)| = 1$, prove that $|f^n(X)| = 1$, where $n = |X|$.
I'm having some difficulty with this question. I understand that I must prove if $f^k$ has image size 1, then $f^n$ also has image size 1 where $n = |X|$, but I'm not completely sure how to start. I was thinking of using mathematical induction for $|X| \geq 1$, is this the right direction? Could anyone give me a general outline of where to go with this question?