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\begin{document}

\title{Homework \#7, Real Analysis I, Fall 2012}


\author{Marius Ionescu}
\date{\bf{Due Friday, 11/2/2012}}
\maketitle
\begin{problem}[Exercise 3.3.1 on page 87]
  Show that if $K$ is compact, then $\sup K$ and $\inf K$
both exist and are elements of $K$.
\end{problem}

\begin{problem}[Exercise 3.3.3 on page 88]
Show that the Cantor set defined in Section~3.1 
is a compact set.
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 3.3.5 on page 88]
Decide which of the following sets are compact. For
those that are not compact, show how Definition~3.3.1 breaks down.  In
other words, give an example of a sequence contained in the given set 
that does not possess a subsequence converging to a limit in the set.
\begin{itemize}
\item[(a)]  $\Q$.
\item[(b)] $\Q\cap[0,1].$
\item[(c)]  $\R$.
\item[(d)]  $\Z\cap[1,10]$.
\item[(e)]  $\{1, 1/2, 1/3, 1/4, 1/5,\dots\}$.
\item[(f)]  $\{1, 1/2, 2/3, 3/4, 4/5,\dots\}$.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 4.2.1 on page 108] Use Definition 4.2.1 to
  supply a proof for the following statements.
\begin{itemize}
\item[(a)]  $\lim_{x\to 2}(2x+4)=8$.
\item[(b)] $\lim_{x\to 0}x^3=0$.
\item[(c)] $\lim_{x\to 2}x^3=8$.
\item[(d)] $\bigl[[x]\bigr]=3$, where $\bigl[[x]\bigr]$ denotes the
  greatest integer less than or equal to $x$.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 4.2.2 on page 108] Assume a particular $\delta>0$ has been constructed as
  a suitable response to a particular $\varepsilon$ challenge. Then,
  any larger/smaller (\emph{pick one}) $\delta$ will also
  suffice. (\textbf{Please justify your answer}).
\end{problem}
\begin{problem}[Exercise 4.2.6 on page 108]
  Let $g:A\to \R$ and assume that $f$ is a bounded function on $A$
  (i.e., there exists $M>0$ satisfying $\vert f(x)\vert \le M$ for
  all $x\in A$). Show that if $\lim_{x\to c}g(x)=0$ then $\lim_{x\to c}g(x)f(x)=0$.
\end{problem}
\end{document}
