Convergence theorem for uniformly integrable martingales

Question

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. $$ X_0 =\mathbf{E}[X_\infty| \mathcal{F}_0] \; ?$$