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\title{Orders}
\date{Due January 19, 2022}

\begin{document}
\ \vspace{-.25in}

\maketitle

\vspace{-.15in}

\prob{
Let $S$ be an ordered set and let $\emptyset \subsetneq A \subseteq B \subseteq S$.
Suppose that $A$ and $B$ both have both an infimum and a supremum.
Prove that $\inf B \leq \inf A \leq \sup A \leq \sup B$. 
}

\vfill

\prob{
Let $S$ be an ordered set and let $A \subseteq S$ be a nonempty and finite. 
Prove (using induction) that $\inf A$ and $\sup A$ both exist and are both in the set $A$.
}

\vfill


\end{document}