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\title{MAT 108 PS08}
\author{Melissa Zhang}
\date{Due \textbf{Tuesday}, May 26, 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 Practice working with the rigorous definitions of `limit',  `convergence', and related terms.
	\item Gain familiarity working with rational and irrational numbers, in preparation for the next unit on cardinality. 
\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 if a sequence of real numbers $(x_k)$ converges, then the limit is unique. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $(a_k)$ be a sequence of real numbers that converges to $A \neq 0$. Prove that 
\[
    \lim_{k \to \infty} \frac{1}{a_k} = \frac{1}{A}.
\]


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

In class, we proved the half of the Monotone Convergence Theorem for increasing sequences (Theorem 10.19). Prove the analogous statement for decreasing sequences, either by modifying the proof for Theorem 10.19 or by directly using Theorem 10.19.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% diverge
In class we proved that the sequence $(-1)^k$ diverges; it is bounded but not monotone. In this exercise, we consider sequences that are monotone but not bounded. For concreteness, we focus on increasing sequences.


We think of a positive real number as being ``close to infinity'' if it is ``far from 0''. 
For example, the sequence $(x_k)$ where $x_k = k$ approaches infinity because no matter how big a number $M$ you give me, I can find an $x_N$ such that all terms afterwards are greater than $M$. 

\bea

\item Use the intuition above to write down a rigorous mathematical definition for $\lim_{k\to\infty}x_k = +\infty$. 
Your definition should start with the following phrase:

\begin{quote}
We say that a sequence $(x_k)$ \emph{approaches positive infinity} and write $\lim_{k\to \infty} x_k = +\infty$ if ...
\end{quote}

\item Prove that $\lim_{k \to \infty} k^2 = +\infty$.

\item Prove that the sequence $x_k = (-1)^k k$ does \emph{not} approach positive infinity.
\ee




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

Suppose $r,s \in \N$ are both \emph{not} perfect squares. Prove that $\sqrt{r} + \sqrt{s}$ is irrational. 





 

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