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.
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.
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 $a\notin S$, $S\neq E$.