Using the definition that X is a sub-exponential rv if $E(e^{sX})\le \exp({s^2v^2/2})\ \forall s:|s|\le 1/b$ for some $b,v>0$, and the assumption that $E(X^2)<\infty$ ,I need show 2 things:
I have shown the first part using jensen's inequality. I can't get a hold on how to do the second part.
$$\frac {e^{tx}-1-tx} {t^{2}} \to x^{2}/2$$ as $ t\to 0$. By Fatou's Lemma we get $$EX^{2}$$ $$ \leq 2\lim \inf E\frac {e^{tX}-1-tX} {t^{2}}$$ $$\leq 2\lim \inf\frac {e^{t^{2}v^{2}/2}-1} {t^{2}} =v^{2}.$$
[I have used the fact that $e^{y} \geq 1+y$ for al $y \in \mathbb R$].
After proving $E(X)=0$, expand the $\exp(x)$ into $1+x+x^2/2!+\cdots$ stuff on both sides and divide by $s^2$ . Then take limit $s\to0$.