but why is this has to be in Z though?
Now, it appears that you are introducing a new question.
Consider the intervals [0, 1) and [0, 1] as subsets of ℝ
Both are bounded above. Both have the same least upper-bound (supremum).
One of these intervals, however, does not have a largest element. Indeed - to prove this, you could take and arbitary element and find one that is larger, and still in the interval.