Let $ f_n\colon [0,\infty) \to \mathbb{R} $ be a sequence of functions and let $ g\colon [0,\infty) \to \mathbb{R} $ be such that $ \left| {f_n \left( x \right)} \right| \leqslant \left| {g\left( x \right)} \right|\, $ for every $x$ and $n$. Suppose in addition that $ \int\limits_0^\infty \! {f_n } \left( x \right) \, dx$ and $\int\limits_0^\infty \! {g\left( x \right) \, dx} $ exist.
It's true that if $ f_n \to 0 $ pointwise, then $ \int\limits_0^\infty f_n\, dx \to 0 $?
This is a calculus course. When we say integrable, I mean in the Riemann sense. I don't know anything about the Lebesgue integral, and I can't use it.