Fatou Lemma: Why is $\lim\inf f_n = 0$ where $f_n = \chi_{[n,n+1]}$

Question

In this wildly popular post, there is a claim:

I like to remember this by example; specifically let $f_n = \chi_{[n,n+1]}$. Then $\lim \inf f_n = 0$, and $\lim \inf \int f_n = 1$.

So $f_n = \chi_{[n,n+1]}$ is a family of rectangles. I can appreciate that $\lim \inf \int f_n = 1$ since the area underneath is always $1$ regardless

How do you justify "$\lim \inf f_n = 0$", since $f_n$ maintains a height of $1$ always?

Answer 1

The point-wise limit of $f_n$ as $n$ goes to $\infty$ is $0$. Its limit is a line on the $x$-axis everyhwere except at $\infty$ it is $1$.