\documentclass[12pt]{article}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{fullpage,amsmath,amssymb,graphicx,color}
\usepackage{pifont}% http://ctan.org/pkg/pifont
\newcommand{\cmark}{\ding{51}}%
\newcommand{\xmark}{\ding{55}}%

\usepackage{amsthm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{assumption}[theorem]{{\rm \textbf{Assumption}}}

\def\NN{\mathbb{N}}
\def\ZZ{\mathbb{Z}}
\def\RR{\mathbb{R}}
\def\QQ{\mathbb{Q}}
\def\CC{\mathbb{C}}
\def\KK{\mathbb{K}}


\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cC{\mathcal{C}}
\def\cF{\mathcal{F}}
\def\cH{\mathcal{H}}
\def\cL{\mathcal{L}}
\def\cN{\mathcal{N}}
\def\cO{\mathcal{O}}
\def\cP{\mathcal{P}}
\def\cR{\mathcal{R}}
\def\cS{\mathcal{S}}

\def\ee{\varepsilon}
\def\ii{\mathfrak{i}}

\newcommand{\mb}[1]{\ensuremath{\mathbf{ #1 }}}
\newcommand{\norm}[1]{\ensuremath{\left\| #1\right\| }}
\newcommand{\enorm}[1]{\ensuremath{|\!|\!| #1|\!|\!| }}

\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator*{\supp}{supp}
\DeclareMathOperator*{\Ker}{Ker}
\DeclareMathOperator*{\Range}{Im}
\DeclareMathOperator*{\spn}{span}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{\bf{\Large\bf Section 3.2:} 2 (optional), 3}
\end{center}
\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}


\item[2.] (Optional) Let $\Omega$ be a bounded open set, and let
  $p\in[1,\infty)$.  Show that the space
  $(C(\overline{\Omega}),\|\cdot\|_{L^p(\Omega)})$ is not complete.
    Hint: Construct a Cauchy sequence that does not converge. 

\item[3.] Suppose that the space $\cP$ of all polynomials
  $p:\RR\to\RR$ is equipped with the norm
  $\|p\|\doteq\sup_{x\in[0,1]}|p(x)|$.  Show that $(\cP,\|\cdot\|)$ is
  not complete.
   \begin{quote}
    {\bf Hint:}  You might want to consider $f(x)=e^x$ and its (Maclaurin series) partial
    sum approximations.  Dini's Theorem (Exercise 2.4.1) may be
    relevant in such an argument.
  \end{quote}
\end{enumerate}


\end{document}