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\begin{document}
\begin{center}
Homework 5
\end{center}

\noindent
Due Date: March 12, 2013.\\

\noindent
Page 160 of Rosenlicht: 1, 2, 3, 4, 20, 35.\\

\begin{center}
Additional Problems:
\end{center}

\noindent
1. Let $f_k$ be a sequence of bounded Riemann integrable functions defined on $[a,b]$.  Suppose that $f_k\to f$ uniformly.  Then prove that $f$ is integrable on $[a,b]$ and
$$
\int_{a}^{b} f_k(x)dx\to \int_{a}^{b} f(x)dx.
$$

\noindent
2.  Prove that if $\sum_{k} g_k$ is a series of functions that converges uniformly on a set $S$ and if $h$ is a bounded function on $S$, then
$$
\sum_{k} hg_k
$$
converges uniformly on $S$.\\

\noindent
3.  Let $s(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}\cdots$ and $c(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}\cdots$ for all $x\in\R$.
\begin{itemize}
\item[(a)] Prove that $s'=c$ and $c'=-s$.
\item[(b)] Prove that $(s^2+c^2)'=0$.
\item[(c)] Prove that $s^2+c^2=1$.
\end{itemize}

\end{document}











