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\begin{center}
{\huge  \sc  Math 117: Homework 6} \\
{\large  Due Monday, May 19th 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*}
 Consider the sequences defined as follows:
\[ a_n = (-1)^{n+1} , \quad b_n = - \frac{1}{n}, \quad c_n = 2n . \]
\begin{enumerate}[(a)]
\item For each sequence, give an example of a monotone subsequence.
\item For each sequence, give its set of subsequential limits. Justify your answer.
\item For each sequence, give its $\liminf$ and $\limsup$. Justify your answer.
\item Which of the sequences converges? Diverges to $+\infty$? Diverges to $-\infty$? (You do not need to justify your answer.)
\item Which of the sequences is bounded? (You do not need to justify your answer.)
\end{enumerate}


\section*{Question 2}
\begin{enumerate}[(a)]
\item State the definition of convergence for a sequence $s_n$ to a limit $s$.
\item State what it means for a sequence $s_n$ to \emph{not} converge to a limit $s$ by negating the definition from part (a).
\item Suppose that $s_n$ does \emph{not} converge to $s \in \mathbb{R}$. Prove that there exists $\epsilon >0$ and a subsequence $s_{n_k}$ so that $|s_{n_k} - s| \geq \epsilon$ for all $k$.
\end{enumerate}



\section*{Question 3*}  
One can show that the set of rational numbers $\mathbb{Q}$ can be listed as a sequence $r_n$. The exact procedure is a little tedious, but you can get an idea of how it works by considering the below diagram from the textbook.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{RationalsDiagram.pdf}
\end{center}
\end{figure}
For example, $r_1 = 0, r_2 = 1, r_3 =1/2$, and so on. Note that some numbers, such as $-1$, are included multiples times.


\begin{enumerate}[(a)]
\item For any $\epsilon >0$ and $a \in \mathbb{R}$, show that the set $\{r \in \mathbb{Q} : |r-a|< \epsilon \}$ contains infinitely many elements. 
\item Let $r_n$ be the sequence of rational numbers. Use part (a) to show that for any $a \in \mathbb{R}$, there exists a subsequence $r_{n_k}$ that converges to $a$. 
\item Let $r_n$ be the sequence of rational numbers. Show that there exists a subsequence $r_{n_k}$ satisfying $\lim_{k \to +\infty} r_{n_k} = +\infty$.
\end{enumerate}



\section*{Background on Infinite Series}
In calculus, you encountered infinite series of the form 
\[ \sum_{k=1}^\infty a_k = a_1 + a_2 + a_3 + \dots . \]
In fact, these are just limits of sequences. In particular, if we define the sequence 
 \[ s_n = \sum_{k=1}^n a_k = a_1 + a_2 + \dots + a_n\]
  to be the sum of the first $n$ terms of the series, then we define the value of the infinite series to be
  \[ \sum_{k=1}^\infty a_k = \lim_{n \to +\infty} s_n . \]
\begin{defi}
Given a series $\sum_{k=1}^\infty a_k$, define the sequence $s_n = \sum_{k=1}^n a_k$. Then the series  $\sum_{k=1}^\infty a_k$ \emph{converges} to a number $L$ if and only if the sequence $s_n$ converges to $L$. Likewise, the series \emph{diverges to $+ \infty$ or $-\infty$} if and only if the sequence $s_n$ diverges to $+ \infty$ or $- \infty$.\end{defi}


 
\section*{Question 4* (Cauchy criterion)} 
Recall that a sequence  $s_n$ is a \emph{Cauchy sequence} if 
\[ \text{ for all } \epsilon >0, \text{ there exists } N \in \mathbb{R} \text{ so that } n , m > N \text{ ensures } |s_n - s_{m}|<\epsilon . \]
\begin{enumerate}[(a)]
\item Prove that the following is an equivalent definition of a Cauchy sequence:
\[ s_n \text{ is a Cauchy sequence if, for all } \epsilon >0, \text{ there exists } N \in \mathbb{R} \text{ so that } n > m > N \text{ ensures } |s_n - s_{m}|<\epsilon . \] 
\item Prove the following theorem about series, known as the Cauchy criterion.
\begin{thm}[Cauchy Criterion]
A series $\sum_{k=1}^{\infty} a_k$ is convergent if and only if 
\[ \text{ for all } \epsilon >0 \text{ there exists } N \in \mathbb{R} \text{ so that } n > m > N \text{ ensures } \left| \sum_{k={m+1}}^n a_k  \right| < \epsilon . \]
\end{thm}
\item Now use Theorem 1 to prove the following corollary:
\begin{cor}
If a series $\sum_{k=1}^{\infty} a_k$ is convergent, then $\lim_{k \to +\infty} a_k = 0$.
\end{cor}
\end{enumerate}



\section*{Question 5} 
\begin{enumerate}[(a)]
\item Prove the following by induction: for $a \neq 1$,
\[ \sum_{i=0}^{m-1} a^i =1 + a + a^2 + \dots + a^{m-1} = \frac{1-a^m}{1-a} . \]
\item Use part (a) to show that
\[ \sum_{i=n}^{m-1} a^i = a^n + a^{n+1} + \dots + a^{m-2} + a^{m-1} = \frac{a^n - a^m}{1-a}. \]
\item Recall that, by the triangle inequality,
\[ \left| \sum_{i=1}^n a_i \right| = | a_1  + a_2 + \dots + a_n | \leq |a_1|+|a_2| + \dots + |a_n| = \sum_{i=1}^n |a_i| . \]
  Let $s_n$ be a sequence such that $|s_{n+1}-s_n| \leq 4^{-n}$ for all $n \in \mathbb{N}$. Use part (b) and the above inequality to prove $s_n$ is a Cauchy sequence.
\item Does the sequence from part (c) converge? Justify your answer.
\end{enumerate}



\section*{Question 6* (decimal expansions)}  
In this problem you will show that any number that can be represented as a nonnegative decimal expansion can be thought of as   the limit of a bounded increasing sequence of real numbers. Since all bounded monotone sequences converge, this guarantees that any decimal expansion you can imagine represents (converges to) a real number. \\

Suppose we are given a decimal expansion $K.d_1 d_2 d_3 d_4 \dots$, where $K$ is a nonnegative integer and each $d_j \in \{ 0,1,2,3,4,5,6,7,8,9\}$. Let
\[ s_n = K + \frac{d_1}{10^1} + \frac{d_2}{10^2} + \dots + \frac{d_n}{10^n} .\]

\begin{enumerate}[(a)]
\item Show $s_n$ is an increasing sequence. (This is almost obvious. Your proof should be short.)
\item Use the result from Q5(a) to prove that $\frac{9}{10} + \frac{9}{10^2} + \dots + \frac{9}{10^n} = 1 - \frac{1}{10^n}$.
\item Use part (b) to prove that $s_n$ is a bounded sequence. 
\item Since $0.\bar{9} = 0.999 \dots$ and $1$ are both decimal expansions, by what you have shown, they both correspond to a real number. Use the hint from part (b) to show that they actually correspond to the same real number.
\end{enumerate}


 \section*{Question 7 (geometric series)}  
 
\begin{enumerate}[(a)]
\item Prove that for $|r|<1$, $\sum_{k=1}^\infty r^k = \frac{1}{1-r}$.
\item Prove that for $|r| \geq 1$,  $\sum_{k=1}^\infty r^k$ does not converge. (\textbf{Hint}: Use Corollary 2 from Q4.)
\end{enumerate}

 \section*{Question 8*}
Let $s_n$ be a sequence of nonnegative numbers, and for each $n$ define $\sigma_n = \frac{1}{n} (s_1 + s_2 + \dots + s_n)$. 
\begin{enumerate}[(a)]
\item Show $\liminf s_n \leq \liminf \sigma_n \leq \limsup \sigma_n \leq \limsup s_n$. \\
(\textbf{Hint:} For the first inequality, show that $M>N$ implies
\[ \inf \{ \sigma_n : n> M \} \geq \left(1 - \frac{N}{M} \right) \inf \{ s_n : n > N \}. \]
For the last inequality, show first that $M>N$ implies 
\[ \sup \{ \sigma_n : n>M \} \leq \frac{1}{M} (s_1+s_2 + \dots +s_N) + \sup \{ s_n : n >N \}.) \]
\item Show that if $\lim s_n $ exists, then $\lim \sigma_n$ exists and $\lim \sigma_n = \lim s_n$.
\item Give an example for which $\lim \sigma_n$ exists but $\lim s_n$ does not exist.
\end{enumerate}

\section*{Question 9}
Suppose $s_n$ and $t_n$ are bounded sequences.
\begin{enumerate}[(a)]
\item Prove that $\limsup s_n + t_n \leq \limsup s_n + \limsup t_n$.
\item Give an examples of bounded sequences $s_n$ and $t_n$ for which $\limsup s_n +t_n < \limsup s_n + \limsup t_n$.
\end{enumerate}



\section*{Question 10*}
Let $s_n$ be the sequence defined in the following figure from the textbook:
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.4\textwidth]{HW6Figure.pdf}
\end{center}
\end{figure}
\begin{enumerate}[(a)]
\item Find the set $S$ of subsequential limits of $s_n$. Justify your answer.
\item Determine $\limsup s_n$ and $\liminf s_n$. Justify your answer.
\end{enumerate}


\section*{Question 11}
\begin{enumerate}[(a)]
\item Suppose $s_n$ has a subsequence $s_{n_k}$ that is bounded. Show that this implies $s_n$ has a convergent subsequence.
\item Suppose that $s_n$ has no convergent subsequences. Prove that $\lim_{n \to +\infty} |s_n| =+\infty$. 

(Hint: prove the result by contradiction, by showing that if $\lim_{n \to +\infty} |s_n| \neq +\infty$, then $s_n$ has a bounded subsequence.)
\end{enumerate}

\section*{Question 12*}

In this problem, we will consider sequences $s_n$ satisfying the following property:
\begin{align}
\tag{*} \text{$\exists s \in \mathbb{R}$ s.t. every subsequence $s_{n_k}$ of $s_n$ has a further subsequence $s_{n_{k_l}}$ satisfying $\lim_{l \to +\infty} s_{n_{k_l}} = s$.}
\end{align}
\begin{enumerate}[(a)]
\item Prove that if $\lim s_n = s$, then property (*) holds.
\item Prove that if property (*) holds, then $\lim s_n = s$. ({Hint:} Use HW6, Q2, part c)
\end{enumerate}

 

 
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