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\title{MTH435 - Homework 8}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 11/29/17.
 
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Let $A \subseteq \R$, $f,g \colon A \to \R$ and $c$ a cluster point of $A$. Prove the following.
  \begin{itemize}
   \item[(i)] $\lim_{x \to c} (fg)(x) = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$.
   \item[(ii)] If $\lim_{x \to c} g(x) \neq 0$ then $\lim_{x \to c} \left(\frac{f}{g} \right) (x) = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c}g(x)}$.
  \end{itemize}

 \end{que}

 \begin{que}
  Compute the following limits, if they exist. For an extra challenge, try to find them directly from the definition (no limit theorems!).
  \begin{itemize}
   \item[(i)] $\lim_{x \to 1} \frac{x^3 - 2x^2 - 5x + 6}{x^2 - 1}$ \vskip 0.1cm
   \item[(ii)] $\lim_{x \to 0} |x|^3 \sin \left( \frac{1}{x^2} \right)$ \vskip 0.1cm
   \item[(iii)] $\lim_{x \to 0} \mathrm{sgn}\left( \cos \left( \frac{1}{x} \right) \right)$ \vskip 0.1cm
   \item[(iv)] $\lim_{x \to 0} \cos \left( \mathrm{sgn} \left( \frac{1}{x} \right) \right)$
  \end{itemize}

 \end{que}

 
 \begin{que}
  Let $A \subseteq \R$, $f_i \colon A \to \R$ for $i = 1,\dotsc,k$ and $c$ a cluster point of $c$. Prove that
  \[
   \lim_{x \to c} ( f_1 + f_2 + \dotsb + f_k)(x) = \sum_{i=1}^k \lim_{x \to c} f_i(x).
  \]

 \end{que}

  \begin{que}
  Let $f \colon \R \to \R$ be defined by
  \[
   f(x) = \begin{cases}
           x \quad &\text{if $x \in \Q$}, \\
           0 \quad &\text{if $x \notin \Q$}.
          \end{cases}
  \]
  \begin{itemize}
   \item[(i)] Show $\lim_{x \to 0}f(x)$ exists.
   \item[(ii)] Show that $\lim_{x \to c} f(x)$ does not exist for all $c \neq 0$.
  \end{itemize}

 \end{que}

 \begin{que}
  Let $f \colon \R \to \R$ and $\lim_{x \to 0} f(x) = L$. Define $g(x) = f(ax)$ for some $a \in \R - \{0 \}$. Prove that $\lim_{x \to 0} g(x) = L$.
 \end{que}


 
 
 
 
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  Let $f \colon \R \to \R$ and define $g(x) = f(x - c)$. Prove that $\lim_{x \to c} g(x) = L$ if and only if $\lim_{x \to 0}f(x) = L$. 
 \end{que}
 
 \begin{que}
  Let $f \colon \R \to \R$ be defined by
  \[
   f(x) = \begin{cases}
           2 + \frac{x}{2} \quad &\text{if $x \in \Q$}, \\
           |x| \quad &\text{if $x \notin \Q$}.
          \end{cases}
  \]
At what points $c \in \R$ does $\lim_{x \to c} f(x)$ exist? Prove your answer.
 \end{que}

 
 \begin{que}
  Let $f \colon \R \to \R$ have the property that $f(x + y) = f(x) + f(y)$ for all $x,y \in \R$. Assume $\lim_{x \to 0}f(x)$ exists.
  \begin{itemize}
   \item[(i)] Prove that $\lim_{x \to 0} f(x) = 0$. (Hint: $f(x+x) = 2f(x)$ for all $x \in \R$.)
   \item[(ii)] Prove that there exists $\alpha \in \R$ such that $\lim_{x \to 1} f(x) = \alpha$. (Hint: $f(x) =  f(x-c) + f(c)$ for all $x,c \in \R$.)
   \item[(iii)] Prove that $\lim_{x \to c}f(x) = \alpha c$ for all $c \in \R$.
  \end{itemize}

 \end{que}

 
 
 
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  It was shown in class that one of the consequences of the Completeness Property of $\R$ is that all Cauchy sequences converge. In this question, we will show that if we assume all Cauchy sequences converge \emph{and} $\R$ has the Archimedean Property then every non-empty set in $\R$ which is bounded above has a least upper bound. This will prove that our definition of completeness of a metric space is equivalent to our original definition of completeness in $\R$.\footnote{Unless you assume $\R$ does not satisfy the Archimedean property, which leads to topics such as Non-standard Analysis and $p$-adic Analysis.}
  \begin{itemize}
   \item[(i)] Prove that if $(x_n)$ is an increasing sequence which is bounded above, then it is Cauchy. Deduce that all bounded monotonic sequences converge.
   \item[(ii)]  Let $A$ be a non-empty set which is bounded above. Construct a Cauchy sequence $(a_n)$ in $A$ as follows.
   \begin{itemize}
    \item Let $a_1 \in A$ and $u_1$ be an upper bound of $A$. Define $p_1 = \frac{1}{2}(a_1 + u_1)$.
    \item If $p_1$ is an upper bound of $A$, define $a_2 = a_1$ and $u_2 = p_1$.
    \item If $p_1$ is not an upper bound of $A$, choose $a_2 \in A$ such that $a_2 > p_1$. Then define $u_2 = u_1$.
    \item Continue inductively, defining $p_{k+1} = \frac{1}{2}(a_k + u_k)$. Deduce that $(a_k)$ are $(u_k)$ are monotonic sequences and thus converge.
   \end{itemize}
   \item[(iii)] Prove that $\sup A = \lim_{n \to \infty} a_n = \lim_{n \to \infty} u_n$.
  \end{itemize}

 \end{que}

 
 
 
 
 
 
 
 
 
 \end{document}