# stationary subset of $cf(\kappa)$

Let $\kappa$ a infinite cardinal with uncountable cofinality and $S\subset\kappa$. We can find a normal function $f$ from $\operatorname{cf}(\kappa)$ on $\kappa$ such that $\sup(\operatorname{rg}(f))=\kappa$. I try to prove the equivalence :

$S$ stationary in $\kappa$ iff $\{\xi<\operatorname{cf}\kappa : f(\xi)\in S\}$ is stationary in $cf\kappa$

$(\Rightarrow)$ : it's ok by supposing the contrary. I can find a club $Y$ in $\kappa$ that doesn't intersect $S$.

$(\Leftarrow)$ : can somebody give me an indication ? it seems to be evident but ...

Thanks.

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This basically boils down to showing that a set $C$ is club iff its preimage $f^{-1}[C]$ under a normal function is club. – Miha Habič Mar 16 '13 at 12:35
Yes, but why $f^{-1}[C]\neq\emptyset$ ? – Marc Moretti Mar 16 '13 at 14:40
Since $f$ is normal, its range is a club, so the intersection of the range and $C$ is a club. Using this you can assume without loss that $C$ is contained in the range of $f$. – Miha Habič Mar 16 '13 at 15:57
ok, thanks a lot. – Marc Moretti Mar 16 '13 at 17:38