> 
> Suppose that $X$ is Hausdorff as well as compact, and let $f : X → X$ be a continuous
>  map. Let $X\_0 = X, X\_1 = f(X\_0)$ and inductively define $X\_{n+1} = f(X\_n)$ for $n ≥ 1$. Show that
>  $A =\bigcap\_n X\_n$ is non-empty.
> 
> 
> 

Seeing as $X$ is Hausdorff and compact, any subset is closed. Hence, $X\_i$ is compact and closed $\forall i\in\mathbb{N}$.

If I could show that the intersection of finitely many of these subsets is always non-zero, then it would follow that the intersection of all of them is non-zero (as $X$ is compact).

Any hints?