I have a general question about expected value.
If $C>0$ is a constant, and $(X_n)$ is a sequence of variables with $X_n<C$ for every $n\ge 1$, is it true that $E[X_n]\le C$ for every $n\ge 1$?
And similarly, if Y is a random variable, and $(X_n)$ is a sequence of variables with $X_n<Y$ for every $n\ge 1$, is it true that $E[X_n]\le E[Y]$ for every $n\ge 1$?
By the definition of the expectation, if the density exists, then
$$E[X_n]=\int_{-\infty}^Cxf_X(x)\ dx\leq C\int_{-\infty}^{\infty}f(x)\ dx=C$$
because the support if the density $\subset (-\infty,C).$
If the density does not exist then the argumentation is the same but it starts with
$$E[X]=\int_{\infty}^C x\ dF_X.$$
For the second question, if $Y<X$ then $$E[X]<E[Y]$$ because every realizations of $X$ and $Y$ are in the same relationship.
The measure theoretical definition of the expectation, when the common probability space $[\Omega,\mathscr A,P]$ is given is
$$\int_{\Omega}X(\omega)\ dP.$$
The result is obvious if you take into account that the measurable function $X$ is changed to $C$ or a pointwise greater $Y$.