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\title{MAT 108 PS06}
\author{Melissa Zhang}
\date{Due Monday, May 11, 2026 at 9:00 pm on Gradescope}

\begin{document}

\maketitle


\section*{Instructions}

There are three goals for this problem set:
\begin{itemize}
    \item Maintain good mathematical writing form, as always.
    \item Review the types of arguments we made back in Chapter 1, but now for the real numbers $\R$ rather than the integers $\Z$.
    \item Practice working with the concepts of maximum, minimum, supremum, and infimum by proving some fundamental properties we discussed in class.
\end{itemize}


\noindent Here are some reminders about best practices:
\begin{itemize}
    \item You absolutely must write in full, connected English sentences. Whenever possible, do not start a sentence with mathematical symbols. Do not use symbols like $\implies$, $\forall$, $\exists$, $\therefore$, etc.\ (except when explicitly discussing statements about logic). 
    \item Your solution must be \emph{neat}. You \textbf{will} be graded on style, which includes  mathematical style and professionalism. 
    \item It is your responsibility to mark where your solutions for each problem begins on Gradescope so that the TA and reader can properly grade your solutions. 
\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Prove that for all $x \in \R$, $x^2 < x^3$ if and only if $x >1$. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Suppose $A \subseteq B \subseteq \R$. 

\bea 
\item Assuming the Completeness Axiom (as we always do), prove that if $B$ is bounded above, then $\sup(A) \leq \sup(B)$. 
\item Prove that if $B$ is bounded below, then $\inf(A) \geq \inf(B)$. (First prove Proposition 8.53 using the Completeness Axiom.)
\ee


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $A \subseteq \R$. Prove that if $\max(A)$ exists, then $\max(A) = \sup(A)$. 

\begin{remark}
In particular, $\sup(A)$ exists, and $\sup(A) \in A$. You can also use a very similar argument to prove the analogous statement for $\min(A)$ and $\inf(A)$.
\end{remark}







\end{document}