A very simple question, but I am not sure for this moment.
I have a strict inequality $a < b$. And I prove that $b = \infty$, say $b$ is an integral. Does this prove that $a < \infty$, that is, $a$ is finite? Question seems simple but things can easily be messed up with infinity.
To be more clear, say $a = \int f dx$ and $b = \int g dx$ and I have $a < b$. If $\int g dx = \infty$, can we say that $\int f dx < \infty$?
If one can construct pathological counterexamples involve limits etc, I will be very happy.
Talking about $\int fdx$ makes no sense, as that is a concrete function, and not a number.
You might want to talk about the definite integral, $\int_0^\infty fdx$ and $\int_0^\infty gdx$. You may also want to discern between showing $\int_0^\infty fdx<\int_0^\infty gdx$ (in which case your claim would be correct), and the case where $f(x)<g(x)$ for all $x$ in the domain over which you are integrating.
In the latter case take $f(x)=1$ and $g(x)=2$.