I am trying to find: $$\lim_{n\to\infty}\inf \ f_n$$

where $f_n = \mathbb{1}_{[n,n+1]}$ is the indicator function that takes value $1$ in the set $[n,n+1]$ and $0$ elsewhere. It seems intuitively obvious to me that the liminf should be zero. However, when I picture the graph of $f_n$, it seems that for $n \geq 1$, it is just a horizontal line at $1$ that goes off to infinity. Can anyone tell me what I am missing here? Thanks!


migrated from stats.stackexchange.com Nov 25 '15 at 8:06

This question came from our site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

  • $\begingroup$ The graph of $f_n$ does not go off to infinity. It hovers above an interval of lenght $1$, and then it is zero everywhere else. $\endgroup$ – RustyStatistician Nov 25 '15 at 6:37
  • $\begingroup$ Please use \liminf_{n\to\infty} or, even better, \liminf\limits_{n\to\infty}, but not \lim_{n\to\infty}\inf. $\endgroup$ – Did Nov 25 '15 at 8:58

Hint: For any fixed $x$, there exists $N \in \mathbb{N}$ such that $N > x$ so that $I_{[n, n + 1]}(x) = 0$ for all $n > N$. So in fact the statement can be strengthened as $$\lim_{n \to \infty} f_n(x) = 0,$$ which of course implies that $\liminf_{n \to \infty} f_n(x) = 0$.

A more general way to see it. Let $A_n$ a sequence of subsets of $X$. Define $$\liminf_{n\to\infty} A_n:=\bigcup_{n\in\mathbb N}\bigcap_{m\ge n} A_m\\\limsup_{n\to\infty} A_n:=\bigcap_{n\in\mathbb N}\bigcup_{m\le n} A_m$$

It holds the following caracterization:

$$\liminf_{n\to\infty}A_n=\{x\in X\,:\,x\in A_n\text{ for all }n\text{ sufficiently large}\}\\ \limsup_{n\to\infty}A_n=\{x\in X\,:\,x\in A_n\text{ for infinitely many values of }n\}$$

And, not so surprisingly,

$$1_{\liminf_{n\to\infty} A_n}=\liminf_{n\to\infty}\ 1_{A_n}\\ 1_{\limsup_{n\to\infty} A_n}=\limsup_{n\to\infty}\ 1_{A_n}$$

This should help you solve it, and maybe provide you with some insight on the problem.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.