We consider a sequence $\{f_n\}$ of measurable functions where $f_n : A \subseteq \mathbb R \to \mathbb R$. By the following post, we have that the set $\{x \mid \lim_{n \to \infty} f_n(x) \text{ exists}\}$ is measurable.
I want to consider a more restricted set, namely, let $S = \{x \in [0,1] \mid \lim_{n \to \infty} f_n(x) \in \mathbb Q\}$. Is $S$ itself measurable? Intuitively it would seem that since we are restricting our initial set of $x$ where $\lim f_n$ exists to a much smaller set, it should not be measurable, how might prove this, or prove that it is measurable instead?