This is the definition of a sub-gassian norm of a random variable $X$: $$ \| X \|_{\Psi_2} = \inf\{ t > 0: \mathbb{E}\exp(X^2/t^2) \leq 2 \}. $$
It is claimed that we have: $$ \mathbb{E}\exp(X^2/\| X \|_{\Psi_2}^2) \leq 2. $$ However, this is not entirely clear to me. In particular, I am trying to show this claim by an approximation argument: Take $t_n$ to be an minimizing sequence and show $\mathbb{E}(X^2/t_n^2) \to \mathbb{E}(X^2/\| X \|_{\Psi_2}^2)$. Now if $\| X \|_{\Psi_2}^2 > 0$, I believe we could use monotone convergence to obtain the result.-Indeed, take $t_n$ to be decreasing wlog and then $\exp(X^2/t_n^2)$ is increasing to $\exp(X^2/\| X \|_{\Psi_2}^2)$. However, how is this claim true when $\| X \|_{\Psi_2}^2 = 0$. Is this even well-defined?