Suppose that
$f(x)$ and $g(x)$ are functions on $[a,\infty)$, and that
$$0 \le f(x) \le g(x) \quad\text{ for all }x \in
[a,\infty).$$Then we say $f$ is bounded between
$0$ and $g$. Observe the following, based on the
inequality above:
 If $\int_a^\infty g(x)\, dx$ converges, then
$\int_a^\infty f(x)\, dx$ must also converge.
 If $\int_a^\infty f(x)\, dx$ diverges, then
$\int_a^\infty g(x)\, dx$ must also diverge.
 If $\int_a^\infty f(x) \,dx$ converges, we know
nothing about $\int_a^\infty g(x) \,dx$  it might
converge or diverge.
 If $\int_a^\infty g(x) \,dx$ diverges, we know
nothing about $\int_a^\infty f(x) \,dx$  it might
converge or diverge.
