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\title{MTH435 - Homework 9}


\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 12/05/18.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Warm-up Questions}
 
 \begin{que}
  Compute the following limits, if they exist, using the definition. If you feel like reliving the good days of calculus, also use the limit laws in the appropriate cases.
  \vspace{-0.1cm}
  \begin{multicols}{2}
  \begin{enumerate}
   \item $\displaystyle \lim_{x \to 1} \frac{x}{x+1}$
   \item $\displaystyle \lim_{x \to +\infty} \frac{\sqrt{x}-2}{\sqrt{x} + 3}$
   \item $\displaystyle \lim_{x \to 1} \frac{x^2 - 3x + 2}{x^2 - 1}$ 
   \item $\displaystyle \lim_{x \to 0} x\mathrm{sgn}(x)$ 
   \item $\displaystyle \lim_{x \to 0} \frac{\sqrt{x+3}}{x}$ 
   \item $\displaystyle \lim_{x \to 0} f(x)$ where\\
   $\displaystyle
    f(x) = \begin{cases}
            1 \quad &\text{if $x \in \Q$},\\
            0 \quad &\text{if $x \in \R - \Q$}.
           \end{cases}$

   
  \end{enumerate}
  \end{multicols}

 \end{que}
 
 \begin{que}
  Let $A \subseteq \R$, $f, g \colon A \to \R$ and $c$ a cluster point of $A$. Suppose that $\displaystyle \lim_{x \to c} f(x) = L$ and $\displaystyle \lim_{x \to c} g(x) = M$ and $b \in \R$. Prove the following. For a bit more of a challenge, try to prove them directly from the definitions.
  \vspace{-0.33cm}
  \begin{multicols}{2}
  \begin{enumerate}
   \item $\displaystyle \lim_{x \to c} (f+g)(x) = L + M$.
   \item $\displaystyle \lim_{x \to c} (fg)(x) = LM$.
   \item $\displaystyle \lim_{x \to c} (bf)(x) = bL$.
   \item If $M \neq 0$ and $g(x) \neq 0$ for all $x \in A$ then $\displaystyle \lim_{x \to c} \left( \frac{f}{g} \right)(x) = \frac{L}{M}$.
  \end{enumerate}
  \end{multicols}
  \vspace{-0.25cm}
 Also prove that \[\lim_{x \to c} a_nx^n + a_{n-1}x^{n-1} + \dotsb + a_1 x + a_0 = a_n c^n + a_{n-1}c^{n-1} + \dotsb + a_1 c + a_0\] 
 \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}
  $A \subseteq \R$. Prove the following. 
  \begin{enumerate}
   \item Let $f,g,h \colon A \to \R$ and $c$ be a cluster point of $A$. If $f(x) \leq g(x) \leq h(x)$ for all $x \in A - \{c \}$ and if $\displaystyle \lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L$, then $\displaystyle \lim_{x \to c} g(x) = L$. 
  \item Let $f \colon A \to \R$ and $c$ a cluster point of $A \cap (c,\infty)$. Then the following are equivalent.
   \begin{enumerate}
   \item $\displaystyle \lim_{x \to c^+} f(x) = L$
   \item For all sequences $(x_n)$ in $A \cap (c,\infty)$ with $x_n \to c$ we have $f(x_n) \to L$.
  \end{enumerate}
\end{enumerate}
 \end{que}

 
 
 
\begin{que}
  Let $f \colon \R \to \R$ and suppose $\lim_{x \to 0} f(x) = L$. Define $g(x) = f(x - c)$. Prove that $\lim_{x \to c} g(x) = L$. 
 \end{que}

 
 
\begin{que}
 Let $A \subseteq \R$, $f \colon A \to \R$ and $c$ a cluster point of both $A \cap (c,\infty)$ and $A \cap (-\infty,c)$. Prove that $\displaystyle \lim_{x \to c} f(x) = L$ if and only if $\displaystyle \lim_{x \to c^-} f(x) = L = \lim_{x \to c^+} f(x)$.
 
\end{que}

 \begin{que}
  Let $A \subseteq \R$, $f \colon A \to \R$ and $c$ a cluster point of $A$. Supose that $\displaystyle \lim_{x \to c} f(x) = L > 0$ and $\displaystyle \lim_{x \to c} g(x) = + \infty$. Prove that $\displaystyle \lim_{x \to c} f(x) g(x) = +\infty$. What happens if $L = 0$? 
 \end{que}

 \begin{que}
 Suppose $f \colon (0,\infty) \to \R$. Show that $\displaystyle \lim_{x \to \infty}f(x) = + \infty$ if and only if $\displaystyle \lim_{x \to 0^+} f \left( \tfrac{1}{x} \right) = \infty$.
\end{que}
 
 
 
 
 
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 \section{Submitted Questions}
 
 \begin{que}
  let $f \colon \R \to \R$ be defined by
  \[
   f(x) = \begin{cases}
           x \quad &\text{if $x \in \Q$},\\
            0 \quad &\text{if $x \in \R - \Q$}.
          \end{cases}
  \]
 Find all points $c$ such that $\lim_{x \to c} f(x)$ exists. Also prove that the limit does not exist at the other points.
 \end{que}

 
 
 \begin{que}
  Let $f, g \colon (a, \infty)$ and suppose $\displaystyle \lim_{x \to \infty} f(x) = L$ and $\displaystyle \lim_{x \to \infty} g(x) = \infty$. Prove that $\displaystyle \lim_{x \to \infty} (f \circ g)(x) = L$.
 \end{que}

 
 
 
 
 
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Challenge Questions}
 
 \begin{que}
  \emph{Hausdorff metric. }Let $X = \{ V \subseteq \R \mid V \neq \varnothing, V \text{ is closed and bounded} \}.$ Prove that $d \colon X \times X \to \R$ defined by
  \[
   d(U,V) = \max \{ \sup_{u \in U} \inf_{v \in V} |u-v|, \sup_{v \in V} \inf_{u \in U} |u-v| \} 
  \]
is a metric on $X$.
 \end{que}
 
 \begin{que}
 \emph{The $p$-adic metric}. This problem requires a little number theory. Let $p$ be a prime number (at the first run through, perhaps fix $p=2$ or $p=3$ to see what is going on).
 \begin{enumerate}
  \item Prove that if $n \in \Z$, then $n$ has a unique expansion in base $p$; that is
  \[
   n = \sum_{i=0}^k a_i p^i 
  \]
where $a_i \in \{ 0,1,\dotsc,p-1\}$ for each $i$. In particular, show that for an integer $n$, there exists a unique $r \geq 0$ such that $n = p^r a$, where $p$ does not divide $a$.
\item Deduce that if $0 \neq x \in \Q$, then there exists a unique integer $r$ such that $x = p^r \frac{a}{b}$, where neither $a$ nor $b$ are divisible by $p$.
\item Show that if $x$ is as above, then if we define $|x|_p = p^{-r}$ and $|0|_p = 0$, then $d_p(x,y) = |x-y|_p$ defines a metric on $\Q$.
\item Show that $(\Q,|\cdot|_p)$ is not complete.
\item (Advanced) Following the method of Question C2 of Homework 5, construct a completion $\Q_p$ of $\Q$ in this metric. Observe that this completion is not an ordered field, and it does not satisfy the Archimidean property.
\item Ponder how one would do analysis on such a space.
 \end{enumerate}

 
 \end{que}
 
 
 
 
 
 
 
 \end{document}
