Convergence theorem for uniformly integrable martingales This is a theorem in my textbook: 
Why "for all $n\in\mathbb{N}$" and not "for all $n\in\mathbb{N}_0$"?
What's wrong with setting $n=0$, e.g.
$$ X_0 =\mathbf{E}[X_\infty| \mathcal{F}_0] \; ?$$

As pointed out by Saz, there is nothing wrong with the case $n=0$. All the assertions are true in this case. 
