$A_n \uparrow A \implies 1_{A_n} \uparrow 1_A$ pointwise?

Question

Let $\Omega$ be the set. Let $1_A$ be the characteristic function on a set, i.e. it has value $1$ iff $x \in A$ and $0$ otherwise. Suppose There's a sequence of subsets $(A_n)_{n\geq 1}$ increasing to a subset $A$. Then is this true: $\lim_{n\rightarrow \infty} 1_{A_n}(w) = 1_A(w), \forall w\in\Omega$?

Answer 1

Hint: Split to two cases, $w \in A$ and $w \notin A$. In each case, think in which (if any) of the subsets $A_n$ you can expect to find $w$.