I would like to prove the following implication:
If the characteristic function $\phi$ is differentiable in zero and $X_1\geq0$ a.s., then $E(X_1)=i\phi'(0)<\infty$.
(This is Exercise 15.4.4 iii) of Probability Theory by A. Klenke, 3rd version).
My steps:
If $\phi$ is differentiable in zero, then I could write it as $\phi'(0)=im$, for some $m\in\mathcal{R}$.
Thus, by exercise 15.4.4 ii) of the same book, $(X_1+...+X_n)/n \xrightarrow{n \to \infty } m$ in probability.
By the weak law of large numbers we know that $(X_1+...+X_n)/n \xrightarrow{n \to \infty } E(X_1)$ in probability, where $X_1,X_2,...$ are i.i.d. random variables.
Concluding, by the uniqueness of the limit, $\phi'(0)=im=iE(X_1)$.
I am not sure about my use of the uniqueness of the limit and the fact that I didn't use the non negativity of the random variable.
What am I doing wrong?
Thank you.