Let $Y$ be compact and $f: Y\to Y$ continuous. Prove that there exists a non empty closed set $A\subseteq Y$ such that $A=f(A)$.
I saw this problem in James Dugundji topology book. Can anyone give some(tiny) hint to solve this problem?
Edit: I don’t know how to solve this “kind” of problem, showing two sets are equal, $A=f(A)$. First(& only) thing coming to mind is $f(Y)$ is compact, $A$ is compact since $A$ is closed in $Y$, $f(A)$ is also compact.