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\lhead{\bf Advanced Calculus (MAA 4212)}
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\rhead{Due Wednesday 2/27}
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%\bigskip\noindent\emph{Reminder: Midterm 2 is on Wednesday 3/20.}

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\begin{enumerate}

\Problem Prove that if $f\st [0,1]\to\mathbb{R}$ is bounded and the lower integral of $f$ is positive then there is an open interval on which $f>0$.

\Problem Define the functions $f_n\st[0,1]\to\mathbb{R}$ by
\[
	f(x)
	=
	\left\{\begin{array}{ll}
		n&\textrm{if $0<x\le 1/n$, and}\\
		0&\textrm{otherwise (if $x=0$ or $x>1/n$).}
	\end{array}\right.
\]
Explain why each $f_n$ is Reimann integrable, the sequence $(f_n)$ converges pointwise to a Riemann integrable function, but
\[
	\lim_{n\to\infty}\int_0^1 f_n\, dx
	\neq
	\int_0^1 f\, dx,
\]
even though the limit on the left hand side exists.

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