I am reading "Topology 2nd Edition" by James R. Munkres.
The following exercise is the exercise 9(a) in section 4.
Show that every nonempty subset of $\mathbb{Z}$ that is bounded above has a largest element.
I solved this exercise, but I am not sure my solution is right.
My solution is here:
Let $\emptyset\neq A\subset\mathbb{Z}$ and suppose that $A$ is bounded above.
Then, $\sup A$ exists by the least upper bound propery of $\mathbb{R}$.
Let $c:=\sup A$.
Assume that $A$ doesn't have a largest element.
Since $c-1$ is not an upper bound of $A$, there is $a\in A$ such that $c-1<a$.
Since $A$ doesn't have a largest element, there exists $b\in A$ such that $a<b$.
Since $a<b<a+1$ doesn't hold, $a+1\leq b$.
So, $c<a+1\leq b$.
So, $c<b$.
Since $c$ is an upper bound of $A$ and $b\in A$, this is a contradiction.
