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\begin{center}
{\huge  \sc  Math 117: Homework 3} \\
{\large  Due Monday, April 21st at 11:59pm}
\end{center}
Questions followed by * are to be turned in. Questions without * are extra practice. At least one extra practice question will appear on each exam.
\section*{Question 1}
Given a sequence $s_n$, suppose that $s, \tilde{s} \in \mathbb{R}$ are both limits of the sequence. Prove that $s = \tilde{s}$. \\

\emph{This shows the limit of a sequence is unique and justifies the fact that we refer to ``the'' limit of a sequence.}

\section*{Question 2}  %new question
Consider $x,y \in \mathbb{R}$ satisfying $x,y \in [1,2]$. Suppose $x^2< 2$ and $y^2 >2$.
\begin{enumerate}[(a)]
\item Suppose $0 < \epsilon < 1$. Prove that $(x+\epsilon)^2 \leq x^2 + 5 \epsilon$ and $(y-\epsilon)^2 \geq y^2 - 4 \epsilon$.
\item Prove that there exists $ \epsilon_1, \epsilon_2 \in(0,1)$ so that $x^2 + 5 \epsilon_1 < 2$ and  $y^2 - 4 \epsilon_2 > 2$.
\item Use parts (a) and (b) to show that there exists $\epsilon_1, \epsilon_2 \in (0,1)$ so that $(x+\epsilon_1)^2 <2$ and $(y- \epsilon_2)^2 > 2$.
\end{enumerate}

\section*{Question 3*} %new questino
Consider an ordered field $F$. For any $a \in F$, recall that $a^2$ is an abbreviation for $a \cdot a$. \\

Fix $a \in F$ with $a \geq 0$. Consider the set $S = \{ c \in F : c \geq 0, \  c^2 \leq a \}$
\begin{enumerate}[(a)]
\item Prove that $S$ is nonempty and bounded above.
\item Suppose  $F = \mathbb{R}$. Explain why, in this case, the supremum of $S$ exists.
\end{enumerate}

\section*{Question 4*}
Consider the set $S = \{ c \in \mathbb{R} : c \geq 0 , \ c^2 \leq 2 \}$. Let $b = \sup(S)$. 
\begin{enumerate}[(a)]
\item Prove that $b \in [1,2]$.
\item  Prove that $b^2 \geq 2$. (Hint: proceed by contradiction, using question 2). 
\item Prove that $b^2 \leq 2$. (Hint: proceed by contradiction, using question 2). 
\end{enumerate}
Combining parts (b) and (c), we see that $b^2 = 2$. \\

In this way, we have shown there exists a real number $b>0$ so that $b^2 = 2$. We can now define the symbol $\sqrt{2}$ by setting $\sqrt{2} := b$. In this way, we have proved $\sqrt{2} \in \mathbb{R}$. Combining this with our result from class that $\sqrt{2} \not \in \mathbb{Q}$, we see that $\sqrt{2} \in \mathbb{I} $, where  $\mathbb{I} := \mathbb{R} \setminus \mathbb{Q}$ is the set if \emph{irrational numbers}.


  
\section*{Question 5*} %formerly practice mditerm Q2

  \begin{enumerate}[(a)]
  \item Prove the following, using the definition of convergence:
\[ \lim_{n \to +\infty} a^n = \begin{cases} 0 &\text{ if } |a|<1. \\
1 &\text{ if } a = 1 . \end{cases} \]
You may use standard facts about the natural logarithm on the real numbers, even though we haven't proved them yet. In particular, you may use that the natural logarithm $\log(x)$ is an increasing function for $x > 0$: that is, for any $x, y >0$,  $x \leq y \iff \log(x) \leq \log(y)$.
\item If $a \leq -1$ prove that the sequence does not converge.
\end{enumerate}
 
  

\section*{Question 6} %formerly practice midterm Q3, I removed part c
Fix $a \in \mathbb{R}$ and consider the collection of rational numbers $S = \{ q \in \mathbb{Q} : a \leq q \}  $.
\begin{enumerate}[(a)]
\item Suppose the underlying field is either $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{Q}$. For which values of $a$ does the minimum of $S$ exist? Justify your answer with a proof.
\item Suppose the underlying field is $\mathbb{F} = \mathbb{R}$. Prove that $\inf(S) = a$.
\end{enumerate}

 
 



\section*{Question 7} %formerly Q5
Given $s, t \in \mathbb{R}$, consider the set  $(s,t] = \{ x \in \mathbb{R}: s< x \leq t \}$. Find the maximum, minimum, supremum, and infimum of the set or state that they do not exist. Justify your answers with proofs.
 
 \section*{Question 8}
 Consider a sequence $s_n$. Prove that $s_n$ is a bounded sequence if and only if $S:= \{ s_n : n \in \mathbb{N} \}$ is a bounded set.



\section*{Question 9* }
\begin{enumerate}[(a)]
\item State the definition of what it means for a sequence $s_n$ to converge to a limit $s$.
\item State the definition of what it means for a sequence $s_n$ to \emph{not} converge to a limit $s$, by negating the definition of convergence.
\item Use the definition of a convergent sequence to prove  that $\lim_{n \to +\infty} \frac{n-3}{n^2+9} = 0$. 
\item Use the definition of a convergent sequence to prove that the sequence  $s_n = (n+1)^2 -2$ does not converge. 
\end{enumerate}

\section*{Question 10*}
Determine if the following sequences converge. Justify your answer with a proof.
\begin{enumerate}[(a)]
\item $a_n = \frac{7n-19}{3n+7}$
\item $b_n = \sin \left( \frac{n \pi}{3} \right)$
\end{enumerate}
You may use standard facts about trigonometric functions, even though we haven't proved them.

\section*{Question 11*}
Consider two sequences, $a_n$ and $b_n$. Suppose $a_n$ converges to $a$ and that there exists $N \in \mathbb{R}$ so that  $b_n = a_n $ for all $n \geq N$. Prove that $b_n$ converges to $a$.
 
  
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