If \(f_i:\mathbb R\to \mathbb R\) are measurable functions, then is \[ \{x:2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \] measurable?
Recall that \(p\) is a limit point of a sequence \(\{a_i\}_{i=1}^{\infty }\) if there exists an increasing sequence of natural numbers \(n_i\) so that \(p=\lim _{i\to \infty }a_{n_i}\).