Supposedly there is a sequence $\{f_n\}$ that converges uniformly to $f$ on $[0, \infty)$ but $$ \int_0^\infty f_n(x) \ dx \not \to \int_0^\infty f(x) \ dx . $$ I can't think of any sequence that behaves like this. I've tried the following sequences with the constant 0 function as the limit:
- $f_n(x) = \dfrac{\cos x}{n}$
- $g_n(x) = \dfrac{e^{-x}}{n}$
I've also thought about using something like cosine as the limit function but that seems like "cheating". Any hints or explicit examples?