Let $S=\{5/n \mid n\in\mathbb N\}$. Find $\sup(S)$ and $\inf(S)$.
I can clearly see that $\sup(S)=5$ and $\inf(S)=0$.
I know that the definition of supremum is for all $\gamma <5$, there exists $x\in S$ such that $\gamma\leq x$. I also know that the definition of infimum is for all $\gamma>0$, there exists $x\in S$ such that $\gamma\geq x$.
I am not allowed to use the Archimedian Property (haven't been taught it in class), but I can use the Completeness Axiom which states that every nonempty set that has an upper bound must have a least upper bound. Clearly, S is not empty and 5 is an upper bound for S, so by Completeness Axiom, S must have a supremum.
I'm unsure of how to continue the proof.