$(Rudin,Ch.7,$ Exercise $12)$ Suppose $g$:$(0,)R$ and $f_n$:$(0,)R$ for $nin{Z}_{+}$are functions

satisfying

$(a)g$ and $f_n$ are Riemann$-$integrable $ont,T$ for any

$(b)|f_n|g,$

$(c)f_{n}f$ uniformly $on$ every compact subset $of(0,),$

Prove that

$\underset{n}{lim}{\displaystyle \underset{0}{\overset{}{}}}{f}_n(x)dx={\displaystyle \underset{0}{\overset{}{}}}f(x)dx$