Suppose $ S=\{m\in \mathbb{Z} | y \lt m \}, y \in \mathbb{R}$ . This set is bounded below by $y$. And it is nonempty since Archimedian Property guarrantes the existence of a natural no $n$ which is bigger than $y$. So it has an infimum (Call it $s\_\*$). I need to show that $s\_\*$ is an integer i.e I need to show that $s\_\* \in S$. 

If $S$ doesn't contain any non-negative integer then $s\_\*=0$ if $0 \in S$ or by well ordering property it is a natural no if $0 \not \in S$. 

else consider $T=\{m \lt 0| m \in S\}$. There are only finitely many integers which are greater than $s\_\*$ and less than $0$. I take the minimum of these integers and claim that the minimum element is the infimum