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?