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?

In my opinion, the best solution is contained in a paper by Luxemburg, Arzelà's dominated convergence theorem for the Riemann integral, American Math. Monthly 78 (1971), available here but not for free.
It is very nice to read this paper, and the proof is essentially elementary. Please do not ask me to copy it in my answer, since it takes a few pages :-)
