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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?

Zibo Hong
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  • If $sup(E)$ is not an integer then $sup(E)-(sup(E)-[sup(E)])/2$ is still an upper bound and smaller than $sup(E)$. – orole Feb 08 '18 at 19:45

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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 $sup(E)\in E$.

Mircea
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