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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 10/03/18.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Warm-up Questions}
 
 \begin{que}
  Exercises from class.
  \begin{enumerate}
   \item Suppose $x_n\to x$ and $c \in \R$. Show that $\lim_{n \to \infty} c x_n = cx$.
   \item Suppose $x_n \to x > 0$. Show that eventually $x_n \geq 0$.
  \end{enumerate}

 \end{que}

 
 \begin{que}
 Let $x,y \in \R$. Prove the following identities.
 \begin{enumerate}
  \item $\max\{x,y\} = \frac{1}{2}(x + y + |x-y|)$.
  \item $\min\{x,y\} = \frac{1}{2}(x + y - |x-y|)$.
 \end{enumerate}
 Now suppose $(x_n)$ and $(y_n)$ are convergent sequences and define the sequences $M_n = \max \{x_n, y_n \}$ and $m_n = \min \{x_n,y_n\}$. Prove that $(M_n)$ and $(m_n)$ converge.
\end{que}
 
 \begin{que}
  Use the limit laws to find the limits of the following sequences.
  \begin{enumerate}
   \item $\frac{\sin(\sqrt{n}) + 5}{n}$.
   \item $\frac{3n^2 + 5n + 19}{4n^2 + 16}$.
   \item $\frac{3 + (-1)^n n}{n^2}$.
   \item $\frac{2^{3n}}{3^{2n}}$.
   \item $\sqrt{n^2 + 5n} - n$.
   \item $\sqrt[n]{a^n + b^n}$ where $0 < a < b$ (Hint: Find good \emph{b}read for your sandwich!)
  \end{enumerate}

 \end{que}

 \begin{que}
  Prove or disprove the following for sequences $(x_n)$ and $(y_n)$.
  \begin{enumerate}
   \item If $(x_n + y_n)$ converges then $(x_n)$ and $(y_n)$ converge.
   \item If $(x_n y_n)$ converges and $(x_n)$ converges then $(y_n)$ converges.
   \item If $(x_n y_n)$ converges and $(x_n)$ converges to $x \neq 0$ then $(y_n)$ converges.
   \item If $(x_n)$ converges and $y_n = x_{n+1} - x_n$ then $(y_n)$ converges.
   \item If $y_n =  x_{n+1} - x_n$ converges to $0$ then $(x_n)$ converges.
  \end{enumerate}

 \end{que}

 
 
 %\begin{que}
 % Find the limits of the following sequences.
 % \begin{enumerate}
 %  \item[(a)] $(n!/n^n)$.
 %  \item[(b)] $(\sqrt[n]{x^n + y^n})$ for $0 < x < y$.
 %  \item[(c)] $\left(\left(1-\frac{1}{n}\right)^n\right)$
 % \end{enumerate}
 %\end{que}



 
 
 %\begin{que}
 % Compute $\limsup x_n$ and $\liminf x_n$ for the following sequences.
 % \begin{enumerate}
 %  \item[(a)] $x_n = \sin \left(\frac{n \pi}{2}\right)$.
 %  \item[(b)] $x_n = 1 + \frac{\sin(n)}{n}$.
 %  \item[(c)] $x_n = (-1)^n \left( 1 + \frac{1}{n} \right)$.
 % \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[(a)] $x_n = 1 + \frac{1}{n}$ and $y_n = 1 - \frac{1}{n}$.
% \item[(b)] $x_n = (-1)^n$ and $y_n = (-1)^{n+1}$.
% \item[(c)] $x_n = y_n = (-1)^n$.
%\end{enumerate}
% 
%\end{que}



 
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Submitted Questions}
 
 \begin{que}
  Suppose that for all $\eps > 0$ there exists $K$ such that $n \geq K$ implies $|x_n - y_n| < \eps$. Prove that $(x_n)$ converges if and only if $(y_n)$ converges.
 \end{que}
 
 \begin{que}
  Suppose $x_n \to 0$ and $(y_n)$ is bounded. Prove or disprove that $(x_n y_n)$ is convergent.
 \end{que}


 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Challenge Questions}
  
 
\begin{que}
You may find previous homework sheets useful for this. Let $\mathcal{T} = \{ U \subseteq \R \mid U \text{ is open} \}$.
 \begin{enumerate}
  \item[(T1)] Show that $\R, \varnothing \in \mathcal{T}$.
  \item[(T2)] Show that if $(U_t)_{t \in T}$ is any family of members of $\mathcal{T}$, then $\bigcup_{t \in T} U_t$ is in $\mathcal{T}$.
  \item[(T3)] Show that if $U_1, U_2 \in \mathcal{T}$ then $U_1 \cap U_2 \in \mathcal{T}$.
 \end{enumerate}
Any collection of subsets $\mathcal{T}$ satisfying (T1), (T2) and (T3) is called a \emph{topology} for $\R$.
\end{que}

  \begin{que}
  Prove the following are topologies on $\R$.
  \begin{enumerate}
   \item $\mathcal{T} = \mathcal{P}(\R) = \{ U \mid U \subseteq \R \}$.
   \item $\mathcal{T} = \{ \varnothing, \R \}$.
   \item $\mathcal{T} = \{ U \subseteq \R \mid \R - U \text{ is finite} \} \cup \{ \varnothing \}$.
  \end{enumerate}

 \end{que}
 
 \end{document}
 
 
 

 \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}

 \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}
