I don't know I'm misinterpreting, but I'm a little confused about the definition of bounded. As far as I was aware, a bounded set of real numbers is one that is bounded both below and above. Therefore could an unbounded set of real numbers be one that is either exclusively bounded below or above or one that is both unbounded below and above?
Reason for the question is that I was working on a proof of the statement that a non-sequentially compact set of real numbers either has (i) an unbounded sequence in S or (ii) there is a sequence in S that converges to a point $x_0 \notin S$.
So $S$ could be a bounded set or an unbounded set.
In the case that it is unbounded, I think it seems pretty clear that we can construct an unbounded sequence, but if the sequence converges to a lower or upper bound (assuming the set only has one of them) then we could have that statement (ii) is true.
I thought this was pretty solid until I found a solution online that seemed to indicate that a set of the form $(a, \infty)$ or $(-\infty,a)$ is considered bounded.
So for a set of real numbers, is the only option for an unbounded set $(-\infty, \infty)$? Does this change depending on who you talk to or is the accepted definition?