Prove convergence without Lebesgue theory

W. Rudin has the following exercise, "to convince the reader of the power of Lebesgue integration".

Let $0 \leq f_n \leq 1$ be continuous functions from $[0,1]$ to $\mathbb R$, such that they converge pointwise to $0$. Prove that their integrals converge to $0$, without using any Lebesgue theory.

How to do this?

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You can exchange limit and Riemann integral for uniformly convergent sequences of Riemann integrable functions. – Michael Greinecker Sep 25 '12 at 11:59
@MichaelGreinecker But only pointwise convergence is assumed. – Julián Aguirre Sep 25 '12 at 12:11
Rudin cites two beautiful papers dating back to 1967 and 1970. I have them, and the bounded convergence theorem for Riemann integrals is proved in details. But it is a really technical proof, too long to be copied here. – Siminore Sep 25 '12 at 12:22