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{\bf Question}

Let ${\cal R}[0,1]$ be the set of Riemann-integrable functions
over [0,1].  Find a sequence of functions $\{f_n\}$, $f_n:[0,1]
\rightarrow {\bf R}$, with the properties

\begin{itemize}
\item[i)]
there exists $f:[0,1]\rightarrow {\bf R}$, for all $\ds
x\epsilon[0,1], \,\,\, \lim_{n\to\infty}f_n(x)=f(x)$

\item[ii)]
there exists $M\epsilon{\bf R}$, for all $n\epsilon{\bf N}$, for
all $x\epsilon[0,1], \,\,\, |f_n(x)|\leq M$

\item[iii)]
for all $x\epsilon[0,1] \,\,\, |f(x)|\leq M$

\item[iv)]
for all $n\epsilon{\bf N}, \,\,\, f_n\epsilon{\cal R}[0,1]$

\item[v)]
$f\not\epsilon{\cal R}[0,1]$
\end{itemize}

and summarise these properties in words.

\vspace{0.25in}

{\bf Answer}

Let $r_1, \, r_2 \cdots$ be an enumeration of the rationals in
$[0,1]$.

Let $f_n(x)=\left\{\begin{array}{cl} 1 & {\rm if \ }
x=r_1,\,r_2,\,\cdots r_n\\ 0 & {\rm otherwise}\end{array}\right.$

This provides the required example of a uniformly bounded sequence
of Riemann integrable functions converging to a non-Riemann
integrable function.


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