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\title{MTH435 - Homework 3}
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\begin{document}
 \maketitle
 
 
 
  Solutions to the questions in Section B should be submitted by the start of class on 10/10/18.
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Exercises from class.
  \begin{enumerate}
   \item Show that the following are equivalent for a sequence $(x_n)$.
   \begin{enumerate}
   \item $(x_n)$ does not converge to $x$.
   \item There exists $\eps > 0$ such that for all $k \in \N$ there exists $n_k \geq k$ such that $|x_{n_k} - x | \geq \eps$.
   \item There exists a subsequence $(x_{n_k})$ such that $|x_{n_k} - x| \geq \eps$ for all $k \in \N$.
   \end{enumerate}
   \item Let $(x_n)$ be a bounded sequence and as in class, define $u_n = \sup\{x_m \mid m \geq n\}$ and $v_n = \inf\{x_m \mid m \geq n\}$.
  \begin{enumerate}
  \item Prove that $(u_n)$ is decreasing and $(v_n)$ is increasing.
  \item Prove that $(u_n)$ and $(v_n)$ converge.
  \end{enumerate}
  \end{enumerate}

 \end{que}

 
 \begin{que}
  Give examples of the following.
  \begin{enumerate}
   \item A sequence with infinitely many ceiling terms\footnote{Analogously to floor terms, a term $x_C$ is a ceiling term if $x_C \geq x_n$ for all $n \geq C$.} and infinitely many floor terms.
   \item A sequence with no floor terms and no ceiling terms.
   \item An unbounded sequence which has a convergent subsequence.
  \end{enumerate}

 \end{que}

 
 \begin{que}
 For the following recursive sequences, decide if $x = \lim_{n \to \infty} x_n$ exists and if possible, find $x$. 
 \begin{enumerate}
  \item $x_1 = 100$ and $x_{n+1} = \frac{x_n}{10} + 9$.
  \item $x_1 = 2$ and $x_{n+1} = x_n^2 - 1$.
  \item $x_1 = \frac{5}{2}$ and $x_{n+1} = \frac{1}{5}(x_n^2 + 6)$.
  \item $x_1 = 4$ and $x_{n+1} = 2 + \frac{1}{x_n}$.
  \item $x_1 = 1$ and $x_{n+1} = x_n + \frac{1}{x_n}$.
 \end{enumerate}

 \end{que}
 
 \begin{que}
  Compute $\limsup x_n$ and $\liminf x_n$ for the following sequences.
  \begin{enumerate}
   \item $x_n = \sin \left(\frac{n \pi}{2}\right)$.
   \item $x_n = 1 + \frac{\sin(n)}{n}$.
   \item $x_n = (-1)^n \left( 1 + \frac{1}{n} \right)$.
  \end{enumerate}
 \end{que}
 
 \begin{que} For the following pairs of sequences $(x_n)$ and $(y_n)$, compute $\liminf(x_n) + \liminf(y_n)$, $\liminf(x_n + y_n)$ and $\limsup (x_n) + \liminf (y_n)$.
\begin{enumerate}
 \item $x_n = 1 + \frac{1}{n}$ and $y_n = 1 - \frac{1}{n}$.
 \item $x_n = (-1)^n$ and $y_n = (-1)^{n+1}$.
 \item $x_n = y_n = (-1)^n$.
\end{enumerate}
 
\end{que}
 
 \begin{que}
Let $x_n$ be a bounded sequence. Prove that
 \[
  \liminf(x_n) = -\limsup(-x_n).
 \]  
 \end{que}


 
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  Consider the recursive sequence defined by $x_1 = 11$ and $x_{n+1} = \sqrt{5 + x_n}$. Prove that $(x_n)$ converges and find the limit.
 \end{que}

 
 \begin{que}
  Suppose $(x_n)$ and $(y_n)$ are bounded sequences. Prove that
  \[
   \liminf (x_n) + \liminf(y_n) \leq \liminf(x_n + y_n) \leq \limsup (x_n) + \liminf (y_n).   \]

 \end{que}
 
 
 
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 \section{Challenge Questions}
  
  \begin{que}
  Let $a,b>0$. Let $(x_n)$ be defined recursively by $x_1 = a$ and $x_{n+1} = \sqrt{b + x_n}$. Show that $(x_n)$ converges to $\frac{1+\sqrt{1+4b}}{2}$. \emph{Hint:  To show monotonicity, it may help to prove the equality $(x_{n+1} - x_n )(x_{n+1} + x_n ) = x_n - x_{n-1}$ and compare signs.}
 \end{que}
  
  
  
  
  
\end{document}
