Suppose E ⊆ Z is nonempty and bounded above. Show that sup(E) ∈ E. In particular, sup(E) ∈ Z.

Question

I think that since E is a subset of Z than all the element must be integer.then sup(E) must exist in E. I really do not know how to prove this. could someone help?

Answer 1

It is obvious that $\sup(E) = $ the maximal element of $E$. Another argument: the set $E$ is closed in $\mathbb R$ and bounded above, hence $sup(E)\in E$.