Show that if $S \subset Z$ is bounded above, then $S$ has a maximum, i.e., $\sup S in S$.

Question

Show that if $S \subset Z$ is bounded above, then $S$ has a maximum, i.e., $\sup S \in S$. I don't know how to start or really manipulate this.

Answer 1

Consider $m=\lfloor \sup S\rfloor$ show it is your maximum.

You can start from sup characterisation $\forall \epsilon>0\ ,\exists s\in S\mid s>\sup S-\epsilon$.

Basically since we are dealing with integers $\epsilon=\frac 12$ gives plenty of informations, because for integers $m<n\iff m+1\le n$