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Let $(E,d)$ be a compact metric space. Let $f:E\to E$ be a function such that for all $x,y\in E$ is $d(f(x),f(y))\ge d(x,y).$
- Let $a\in E.$ Prove that $a$ is a limit point of the sequence $(f^n(a))_{n>0}.$ ($f^n(a)$ means the composition of $f,$ $n-$ times. Example: $f^3(a)=f(f(f(a))).$ )
- Conclude that $f(E)$ is dense on $E.$