This is a problem I encountered in an exercise about the application of characteristic function: The first question asks me to prove that the characteristic function of $X - Y$ is real-valued and non-negative when $X$ and $Y$ are i.i.d random variables.
Given this, I have to prove that $X - Y$ being square-integrable implies that $X$ is square-integrable, which in this case is equivalent to show $|\mathbb{E}[X]| < \infty$.
If I am to prove this using characteristic function, I have to show that $\frac{d}{dt} \mathbb{E}[e^{itX}]$ exists, yet I have no idea about how to get this from $\mathbb{E}[(X - Y)^2] < \infty$. Could anyone give me a hint?