I'm having some problem in the computation of limits of indicator functions.
Why the following limit takes values $0$?
$$\lim_{n\to\infty}1_{[n,n+1]}$$
Which are the steps I should follow to compute every kind of limit of such functions? I started thinking about applying the definition of limit as the x-values go to infinity, but at some point I get really confused because of the set of the indicator function… Does it work like the domain for other functions?
How does this work for limsup and liminf of an indicator function?
Thank you!
[Math] Limit of an indicator function
characteristic-functionslimitsreal-analysis
Best Answer
Answering your first question: For a given $x$, there are at most two values of $n$ for which $1_{[n,n+1]}$ is nonzero. As $n\to\infty$, the interval which supports $1_{[n,n+1]}$ will "fly off" to the right, passing any given $x$. So in the tail of the sequence, $1_{[n,n+1]}(x)$ is eventually zero for any give $x$.