Suppose \(f_i:\mathbb {R}\to \mathbb {R}\) is measurable for \(i=1,2,\cdots \). Is the set \[ \{x\in \mathbb {R}\mid 2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \]
measurable? A number \(t\) is a limit point of a sequence if some subsequence converges to it.