$\limsup$ and integral inequality

Question

I have functions $f$, $g$ :$(0, \infty) \rightarrow (0, \infty)$, and $g$ is invertible and decreasing (I don't know if it is relevant or not in this case). I know also that $\limsup_{x \searrow 0} \frac {g(x)} {f(x)} < \infty$. I should be able to deduce that $\int_0^r g(x) dx \leq \int_0^r f(x) dx$ for some small enough $r \in (0, 1)$. It should be easy to see, but the problem is, I don't see how.