Assume that $V$ is a stochastic variable, $g(V)\geq 0$ is a function related to $V$.
If the upper bound of $g(V)$ can be determined, i.e. $g(V)<c$, where $c$ is a constant, does the mathematical expectation inequality $E\{g(V)\}<c$ hold?
I have tried some examples in both continuous case and discrete case, and it seems correct.
If so, how to prove this inequality? If not, is there an example to overthrow it?
yes in general if $X\ge 0$ then $EX\ge 0$, this is a basic property of the expected value called monotonicity or positivity. (in your case we can take $X=c-g(V)$).