Let $g_n=\chi_{(n,n+1)}$ where $\chi$ is the characteristic function of the interval in the subscript. It is given as an example in 4.3 of Real Analysis by Royden and Fitzpatrick that $g_n$ converges pointwise to $g=0$. I don't really understand why though. Is it simply using the below logic? If so, can someone expound on why $\chi_{(\infty,\infty)}=0$. Thanks!
$$ \lim_{n\rightarrow\infty} g_n(x) = \lim_{n\rightarrow\infty} \chi_{(n,n+1)} = \chi_{(\infty,\infty)}=0 $$
Take $x\in\Bbb R$; you want to prove $\lim_{n\to\infty}\chi_{(n,n+1)}(x)=0$. But if $n>x$, then $\chi_{(n,n+1)}(x)=0$. So, the numbers of the sequence $\bigl(\chi_{(n,n+1)}(x)\bigr)_{n\in\Bbb N}$ are all $0$ if $n$ is large enough, and therefore $\lim_{n\to\infty}\chi_{(n,n+1)}(x)=0$.