Another question on stationarity

Question

Let $\kappa < \lambda$ be regular cardinals and let $A \subset \lambda$ be such that each $\alpha \in A$ has cofinality $\gamma < \kappa$. Then the following should hold:

$A$ is stationary iff $ \lbrace$$X \in P_{\kappa} (\lambda) : sup(X) \in A $ $\rbrace$ is stationary in $P_{\kappa} (\lambda)$

The direction from the right to the left is not hard, but despite looking harmless, the $\Rightarrow$ direction puzzles me now for a while.

It should be enough to show that for any club $C$ in $P_{\kappa} (\lambda)$ the set $\underset{\sim}{C}:=$ $\lbrace$ $\beta \in \lambda : \exists X \in C \quad sup(X) = \beta$ $\rbrace$ has a $\gamma$-closed, unbounded subset of $\lambda$ which would imply (by the assumption that each element of $A$ has cofinality $\gamma$) that $\underset{\sim}{C} \cap A \ne \emptyset$, witnessing the stationarity of $ \lbrace$$X \in P_{\kappa} (\lambda) : sup(X) \in A $ $\rbrace$. However all my attempts to show this were cumbersome and dissatisfactory.

Answer 1

Every club subset $C\subset P_\kappa(\lambda)$ contains the collection of size-less-than-$\kappa$ elementary substructures of some first order structure $W=\langle\lambda,\in,\ldots,\rangle$, and the collection of such elementary substructures is club. There is a Skolem function $f:\lambda^{\lt\omega}\to\lambda$ such that any set closed under $f$ is elementary in $W$. The set of ordinals $\beta\lt\lambda$ closed under $f$ is club in $\lambda$, and so there is $\beta\in A$ closed under $f$. Now pick a cofinal subset of $\beta$ and close under $f$ to find an $X$ elementary in $W$ of size less than $\kappa$ with $sup(X)=\beta\in A$. Since $X$ is elementary in $W$, it is in $C$, as desired.