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

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\def\ii{\mathfrak{i}}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
  {\bf {\Large Section 2.4:} 1, 2 (optional)}
\end{center}
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\begin{enumerate}  
\item[1.]  (Dini's Theorem) Let $K$ be a compact metric space, let
  $(f_n)_{n=1}^\infty$ be a sequence in $C(K)$ and $f\in C(K)$.
  Suppose that $f_n\to f$ pointwise, and that $f_n(x)\leq f_m(x)$ for
  all $x\in K$ when $n\leq m$.  Show that $f_n\to f$ uniformly.
\begin{quote}
{\bf Hint:}  Fix $\varepsilon>0$ and define $\cO_n=\{x\in K:\,
f(x)-f_n(x)<\varepsilon\}$.  Argue that these sets are open and nested, $\cO_n\subset\cO_{n+1}$,
for all $n\in \NN$. Argue that pointwise convergence implies
that $\{\cO_n:\,n\in\NN\}$ is an open cover of $K$, and use the
compactness of $K$ accordingly.
\end{quote}

\item[2.] \textbf{(Optional)} Let $\Omega\subset\RR^n$ be a non-empty open subset.  For
  $f\in C(\Omega)$ and compact subsets $K\subset\Omega$, we define
  \begin{align*}
    |f|_K\doteq \sup_{x\in K}|f(x)|~.
  \end{align*}
  For each such $K$, $|\cdot|_K$ is a seminorm on $C(\Omega)$; strict
  positivity fails because there will always be non-zero continuous
  functions that vanish indentically on $K$.
  \begin{enumerate}
  \item Let $(K_i)_{i=1}^\infty$ be a sequence of subsets of $\Omega$
    such that
    \begin{align*}
      K_i\subset \mathrm{int}(K_{i+1})\mbox{ for all }i\geq 1\mbox{
      and }\bigcup_{i=1}^\infty K_i =\Omega~.
    \end{align*}
    Such a sequence of nested sets is guaranteed by Theorem 1.13.6.
    Let $(\alpha_i)_{i=1}^\infty$ be a sequence of positive numbers
    such that $\sum_{i=1}^\infty\alpha_i$ converges.  Now define
    $d:C(\Omega)\times C(\Omega)\to\RR$ by
    \begin{align*}
      d(f,g)\doteq \sum_{i=1}^\infty\frac{\alpha_i|f-g|_{K_i}}{1+|f-g|_{K_i}}~.
    \end{align*}
    Show that $d$ is a metric on $C(\Omega)$.
    
  \item Show that a sequence $(f_m)_{m=1}^\infty$ in $C(\Omega)$
    converges to $f$ in the metric space $(C(\Omega),d)$ if and only
    if $|f_m-f|_K\to 0$ as $j\to\infty$ for each $K$.


  \item Show that the metric vector space $(C(\Omega),d)$ is complete.
  \end{enumerate}
  
\end{enumerate}

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