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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 7} \vs \\
{\bf due Friday 11/1/2013} \vs \vs
\end{center}  

\noindent
{\bf Problem 1.} Suppose $X$ is a metric space, $f_n \in {\mathscr C}(X)$, and $f_n \to f$ uniformly on $X$. \vs
\begin{enumerate}
\item[(i)]
Show that $f_n(p_n) \to f(p)$ in $\CC$ whenever $p_n \to p$ in $X$. \vs 
\item[(ii)]
Show that if each $f_n$ is uniformly continuous on $X$, so is $f$. \vs
\end{enumerate}

\noindent 
{\bf Problem 2.} Consider 
$$
f(x) = \sum_{n=1}^\infty \frac{1}{1+n^2x}.
$$
For what values of $x$ does the series converge? On what intervals does it converge uniformly? Where is $f$ continuous? Is $f$ a bounded function on its domain? \vs

\noindent
{\bf Problem 3.} Suppose $f: [a,b] \to \RR$ is a $C^{\infty}$ function such that the sequence $\{ f^{(n)} \}$ of its successive derivatives converges uniformly on $[a,b]$. Find the limit function.  \vs

\noindent
{\bf Problem 4.} Prove that there is no sequence of continuous functions $f_n: \RR \to \RR$ such that 
$$
\lim_{n \to \infty} f_n(x) = \begin{cases} 0 & \quad \text{if} \ x \in \QQ \\ 1 & \quad \text{if} \ x \notin \QQ. \end{cases} \vs
$$
(Hint: Use Baire's theorem.) \vs

\noindent
{\bf Problem 5.} Fix an integer $d \geq 1$. Let $P_n: \RR \to \RR$ be a sequence of polynomials of degree $\leq d$. If $P_n(x) \to 0$ for every $x \in [a,b]$, show that $P_n \to 0$ uniformly on $[a,b]$. (Suggestion: First try the easier case of $d=2$ and $[a,b]=[0,1]$.) \vs

\noindent
{\bf Problem 6.} 
\begin{enumerate}
\item[(i)]
Let $X$ be a compact metric space. Suppose a sequence of continuous functions $f_n : X \to \RR$ converges monotonically increasing to a continuous function $f: X \to \RR$ (that is, $f_1(x) \leq f_2(x) \leq f_3(x) \leq \cdots$ tend to $f(x)$ for all $x \in X$). Prove that $f_n \to f$ uniformly on $X$. \vs 
\item[(ii)]
Suppose a sequence of increasing functions $f_n : [a,b] \to \RR$ converges pointwise to a continuous function $f: [a,b] \to \RR$. Prove that $f_n \to f$ uniformly on $[a,b]$.
\end{enumerate}
(Hint: For (ii), use uniform continuity of $f$.)



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