TITLE: Convergence of Indicator function.
QUESTION [5 upvotes]: I was going through Egorov's theorem on wikipedia. 
It gives a example why should be  $\mu(A)<\infty $. Sequence of real valued indicator function is taken. It claims that the sequence: 
 $ f_n(x)=1_{[n,n+1]}(x) $ converges pointwise for $n\in N$ and $x\in\Re$.
I am not able to understand how it converges pointwise to $0$ ?

REPLY [6 votes]: For the pointwise convergence, you have to fix $x'\in \mathbb R$ and check that a numerical limit exists: in your case that $f_n(x') \to 0$. Clearly, given $x'$ you have that if $n>x'$ then $x'\notin[n,n+1]$ and hence $f_n(x') = 0$ for all $n>x'$, so that for any $x'$ you have $f_n(x')\to 0 $ - which is the pointwise convergence.