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Let $X$ be Hausdorff and compact, and let $f:X\to X$ be a continuous map. Let $X_0=X$, $X_1=f(X_0)$, $X_2=f(X_1)$, etc. Show that $X_0,\ldots,X_n$ is nested.

I'm trying to prove that for $X_n$ defined by $X_0=X$, $X_1=f(X_0)$, $X_2=f(X_1)$, $X_{n+1}=f(X_n)$, ..., the intersection of all $X_n$ is non-empty.

First, I proved that if $X$ is compact, and $V_n$ is a nested sequence of non-empty closed subset of $X$, the intersection of $V_n$ is non-empty.

Then, I only have to prove that $X_0,\ldots, X_n$ is nested from the fact that $f$ is continuous and $X$ is Hausdorff and compact. This is where I got stuck, though.

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It’s certainly clear that $X_1\subseteq X_0$. Apply $f$ to both sides: $X_2=f[X_1]\subseteq f[X_0]=X_1\;.$ Can you see how to convert this into a proof by induction that $X_{n+1}\subseteq X_n$ for each $n\in\mathbb{N}$?

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    awesome thanks! simple and clever.2011-12-02