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Let $(f_n)_{n=1}^\infty$ be a uniformly-bounded sequence of Riemann-integrable functions on $[a, b]$. If $f_n \to 0$ pointwise, prove $\int_a^b f_n \to 0$.

my attempt

Since each $f_n$ is Riemann integrable on $[a, b]$, it is Lebesgue integrable (hence also measurable) on $[a, b]$ and its Lebesgue integral agrees with its Riemann integral. Then Lebesgue's Dominated Convergence Theorem can be directly applied since the functions are uniformly-bounded and the measure of the set is finite.

question

I wonder how the same result could be proved completely as a Riemann integral without invoking any result from Lebesgue's convergence theorems? Thanks a lot.

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    $\begingroup$ This is Arzela dominant convergence theorem. The proof without Lebesgue integral exists but it is very complicated. $\endgroup$ – xbh Aug 8 '18 at 3:08
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    $\begingroup$ @xbh, That is a nice keyword to begin with! $\endgroup$ – Sangchul Lee Aug 8 '18 at 8:49
  • $\begingroup$ @xbh Awesome! I learned something new today :) Thanks. $\endgroup$ – mkmlp Aug 8 '18 at 13:09
  • $\begingroup$ @michaelshiyu My pleasure ;) $\endgroup$ – xbh Aug 8 '18 at 13:10

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