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Let $E$ be an infinite set and $f:E\rightarrow E$. Then there exists a subset $S$ of $E$ such that $\emptyset\neq S\neq E$ and $f(S)\subset S$.

This seems easy, but I don't have an idea. A hint would be nice.

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    @Stefan Walker: That's good, after all it was called$a$hint, not$a$solution.2011-05-20

1 Answers 1

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Thanks to a hint of user6312 and on his suggestion:

Let $a$ be an element of $E$. If there exists $n\geq 1$ such that $f^n(a)=a$, then define $S=\{a,f(a),f^2(a),\ldots,f^{n-1}(a)\}$. Since $S$ is finite, $S\neq E$. Obviously, $f(S)\subset S$.

If $f^n(a)\neq a$ for all $n\geq 1$, then define $S=\{f^n(a)|n\geq 1\}$. Obviously, $f(S)\subset S$, and since $a\notin S$, $S\neq E$.

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    You can accept your own answer, by the way, and in this case should it so that the question does not show up as "no answer accepted".2011-05-20