Prove that for every function , there is a such that .

Hint: A fixed point theorem may be useful. We may us the Countable Principle of Choice, but not the Axiom of Choice.

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- Mar 8th 2010, 09:30 PMselenne431fixed point theorem
Prove that for every function , there is a such that .

Hint: A fixed point theorem may be useful. We may us the Countable Principle of Choice, but not the Axiom of Choice. - Mar 9th 2010, 04:59 AMemakarov
Are you referring to Knaster–Tarski fixpoint theorem? Since it requires a monotonic function, maybe one can consider . Then is monotonic, so it has a fixpoint : , which implies . And the Countable Principle of Choice is used to show that is a complete lattice. I am not sure about this, but it may be a start...