Let $X$ be a subset of $\omega^{\omega}$, $X^{*}$ is defined as:$$\{y:(\exists x\in X,\exists N <\omega)(\forall n >N x(n)=y(n))\}$$ which consists of all sequences in $\omega^{\omega}$ that eventually coincide with some $x$ in $X$. Show that for any set $A \in \omega^{\omega}$ that is meager(countable union of nowhere dense set), there is a nowhere dense set $C$ that $A \subseteq C^*$ .

Here's how far I understand the problem. For any meager set $A$,$A^*$ is also meager,which can be written as a union of non-decreasing sequence of nowhere dense sets $\{C_n:n<\omega, C_n \subseteq C_{n+1}\}$. But I got stuck on how to show that the induced sequence $\{C^{*}_n:n<\omega\}$ is eventually constant. In other words, there exists $N$ such that $C^*_N = A^*$.


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