Suppose the sample space $S$ is finite. Is it possible to define an unbounded random variable on S? Why or why not?
Let's define a sample space $S=\{1,2,3,4,5,6\}$.
From my textbook definition simply said, a random variable $X$ is a function that maps each one of the elements $s\in S$ to a value in $\mathbb{R}^1$.
This question is essentially asking if I can have $X(s) = \pm \infty$
Is this possible? Does it really come down to whether or not we consider $\pm \infty$ a part of $\mathbb{R}^1$?
Yes, that's precisely the crux. This possibility would seem to be excluded from your own definition, of course.
It is both common and useful to define random variables as having outputs in $\mathbb R \cup \{\infty, -\infty\}$, so I'm curious why they've made that choice. But they have, and it's not unreasonable.
That said, the situation would be different if the sample space had even a countably infinite collection of points. It could be the case that your random variable would map onto $\mathbb Z$, for instance, even if it never mapped directly onto $\pm \infty$. This would still permit an unbounded random variable that had probability $0$ of assuming an infinite value. But that case is not possible here, for reasons that I think you see already.