y Proposition \[char-meagre\], we get $z\in 0^\kappa$ and a sequence $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $$A\subseteq \left\{x\in 0^\kappa\colon \exists_{\beta<\kappa}\forall_{\beta<\gamma<\kappa} \exists_{\xi_{\gamma}\leq\xi<\xi_{\gamma+0}} z(\xi)\neq x(\xi)\right\}.$$  