Prove that $1_{E_n}$ is pointwise convergent

Question

Given that $E_n$ is an increasing sequence of subsets of $\mathbb{R}$ .

What I did is that I took $x\in E_m$ for some $m$, then as $E_n$ is increasing, $\forall n\geq m$ we have $x\in E_n$, then $1_{E_n}(x)=1$ , so $1_{E_n}\rightarrow 1$

But what if $x\notin E_n$ for all $n$ ?

Can someone help please ?

Answer 1

Then $1_{E_n}(x) = 0$ for all $n$ which also converge , $1_{E_n}(x) \to 0$.

So to conclude (with our comments) :

$$1_{E_n}(x) \to 1_{\cup_n E_n(x)}$$