I was reading a paper and I found the following statement:
Since $f(t)$ is non-negative and non-decreasing over $t$, then there must exists $a$, $0\leqslant a\leqslant +\infty$, such that $\lim_{t\to\infty}f(t)=a$.
I tried to understand what does it mean but I am only able to say that it is useless. Am I right? I mean, $a$ could be anything (including $\infty$). So the limit could be anything too.
I think the key point is that they are ruling out the situation where $\lim_{t \rightarrow +\infty} f(t)$ doesn't exist. So the function can't behave like $2 + \sin(t)$, which isn't non-decreasing and fails to have a limit as $t \rightarrow +\infty$.
The fact that the limit is positive seems, to me, like a minor point in comparison.