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Homework Statement
Let [tex](X,\tau)[/tex] be a compact Hausdorff space,
and let [tex]f : X \to X[/tex] be continuous, but not surjective. Prove that
there is a nonempty proper subset [tex]S \subset X[/tex] such that [tex]f(S) =
S[/tex]. [Hint: Consider the subspaces [tex]S_n := f^{\circ n}(X)[/tex] where
[tex]f^{\circ n} := f \circ \cdots \circ f[/tex] ([tex]n[/tex] times)].
Homework Equations
The Attempt at a Solution
If such [tex]S[/tex] exists then [tex]f^{\circ n}(S) = S[/tex]. How should I use this in the proof? I don't have any clue where to start.