Trivial question about limit of an indicator function

Question

I've seen in some places where authors write, say, $1_{[0,\infty)}$ as $\lim_{n\to\infty}1_{[0,n)}$, when dealing with some summation or integration. Would it have made any difference if $[0,n)$ were replaced by $[0,n]$? I am just worrying, that $1_{[0,n]}$ might tend to $1_{[0,\infty]}$ instead of $1_{[0,\infty)}$.

Answer 1

$1_{[0,n]}$ also tends to $1_{[0,\infty)}$ at every point of $\mathbb R$. Proof: If $x < 0$ then $1_{[0,n]}(x)=0=1_{[0,\infty)}$ for all $n$. If $x \geq 0$ then $1_{[0,n]}(x)=1=1_{[0,\infty)}(x)$ whenever $n \geq x$. Hence the limit of $1_{[0,n]}(x)$ is equal to $1_{[0,\infty)}(x)$