Let $(A_n)_{n\geq 1}$ be a sequence of sets. Show that $$\limsup_{n\to\infty} 1_{A_n} - \liminf_{n\to\infty} 1_{A_n} = 1_{\{\limsup_{n}A_n\setminus\liminf_{n}A_n\}}.$$

Exercise 2.5

Let ${(A_{n})}_{n \geq 1}$ be a sequence of sets. Show that

\underset{n \to \infty}{limsup} 1_{A_{n}} - \underset{n \to \infty}{liminf} 1_{A_{n}} = 1_{{\underset{n}{limsup} A_{n} ∖ \underset{n}{liminf} A_{n}}} .