This is also from Kunen, Set Theory, ch. II:
Let $A$ be a set of infinite cardinals such that for each $\lambda$ regular $A\cap\lambda$ is not stationary in $\lambda$. Show that there is an injective function $g$ with domain $A$ such that $\forall\alpha\in A(g(\alpha)<\alpha)$.
I tried using $\lambda=\sup A^+$, took a closed unbounded set $C$ disjoint from $A\cap\lambda=A$ and defined $g(\alpha)=\sup(C\cap\alpha)$ if $\sup(C\cap\alpha)>g(\eta)\forall\eta<\alpha$ and $g(\alpha)=\eta$ if $\sup(C\cap\alpha)=g(\eta),\eta<\alpha$, but then realized it could happen that $\alpha>\eta=\sup(C\cap\beta)=g(\beta),\eta<\beta<\alpha$, so g wouldn't be injective.