Show that if E is a nonempty subset bounded of $Z$, then inf E exists and belongs to E

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Assuming that E is nonempty and bounded subset of $\mathbb{Z}$ then by definition of the completeness axiom E has a finite infium i. since i $\in E$ and $E \subset \mathbb{Z}$. Then i therefore exists and belongs to E. I think I am missing something.