Let A be a nonempty bounded above subset of Z. Prove that A has a largest element.
To prove A has a largest element does it mean to prove A is the least upper bound of Z, if so how do I do this?
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.