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\usepackage{verbatim}

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\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}

% ENVIRONMENTS FOR CITING BOOK STATEMENTS
\newcommand{\prop}[1]{\paragraph{Proposition #1}}
\newcommand{\axm}[1]{\paragraph{Axiom #1}}
\newcommand{\thm}[1]{\paragraph{Theorem #1}}


% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\revisedsolution}{{\color{blue} \textsc{\ \\ Revised Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real nubers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\divides}{\ \vert \ }
\newcommand{\st}{ \ : \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\img}{\mathrm{im}} % image
\newcommand{\into}{\hookrightarrow} % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
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% SOME ENVIRONMENTS

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\newcommand{\exercise}{
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\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
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%%%%
\title{MAT 108: Problem Set 7}
\author{(ADD NAME)}
\date{Due 2/28/23 at 11:59 pm on Canvas}

\begin{document}

\maketitle


\paragraph{Reminders:}
\begin{itemize}
%\item Put your name at the top!
\item Exam 2 is Wednesday, March 1, in class. It will cover all the material we covered in February, including all the material on PS 4--7. 
    \begin{itemize}
        \item To study for this exam, I recommend solving problems from the book, and also making sure you are able to solve previous PS exercises.
        \item Once again, style will be very important. If you lost style points on Exam 1, I urge you to look at the comments on your graded Exam 1 and ask me or Hans if you aren't sure why you lost style points.
        \item Discussion on Tuesday, 2/28 will a review session where you'll have the opportunity to practice solving problems similar in flavor to those on the exam.
    \end{itemize}
\item Your homework submission must be typed up in full sentences, with proper mathematical formatting. Handwritten homework submissions will receive a score of 0. Solutions containing incomplete sentences or poor formatting will lose points.
\item You will receive feedback on PS6 by next Tuesday, 2/28. PS5 revisions are due Friday, 3/3 at 11:59 pm. Underneath your old solution, type 
\vspace{-3mm}
    \begin{verbatim}
        \revisedsolution
    \end{verbatim}
\vspace{-7mm}
and then type your revised solution.
\end{itemize}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

Prove that limits of sequences of real numbers are unique. 

\emph{Hint: In other words, prove that if $(x_k)$ converges to $L$ and to $L'$, then $L=L'$.}

\solution




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

The \textbf{Monotone Convergence Theorem} is a powerful tool in analysis. It states that 
\begin{quote}
    \textit{Every monotonic bounded sequence converges.}
\end{quote}
In class, we proved that every \textit{increasing} bounded sequence converges (Theorem 10.19). 
Prove the analogous statement to Theorem 10.19 for \emph{decreasing} bounded sequences.

\emph{(Write down the precise statement you are proving before you prove it. The Proposition environment has been included in the solution area below.)}

\solution

\begin{proposition}
    % add your precise statement here
\end{proposition}

\begin{proof}
    % write your proof here
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

We can restate the Monotone Convergence Theorem as follows:
\begin{quote}
    \textit{If a sequence is monotone and bounded, then it converges.}
\end{quote}
In this exercise, you will see that monotone, bounded sequences are ``special" within the set of convergence sequences.

\bea

\item Prove the following partial converse to the Monotone Convergence Theorem:
\begin{proposition}
    If a sequence converges, then it is bounded.
\end{proposition}

\item Notice that the partial converse does not conclude that the converging sequence must also be monotone. Give an example of a sequence that converges but is not monotone. \emph{Make sure you prove that your sequence indeed converges and is indeed not monotone!}

\ee


\solution

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise 

\begin{definition}
    An integer $n$ is a \emph{perfect square} if $n=m^2$ for some $m \in \Z$. 
\end{definition}

Prove that if $r \in \N$ is not a perfect square, then $\sqrt{r}$ is irrational.
%
\emph{Hint: Emulate the proof of Proposition 11.10, which states that $\sqrt{2}$ is irrational.}


\solution






\end{document}