Clarification on when is $X_T$ integrable (martingale theory)

Question

I have the following homework problem, and I have already done the (a)-part. I am confused as to what condition (b) actually means: does it mean that

1. there exists $C$ such that whenever $T(\omega)\geq n$, we have $|X_n(\omega)|\leq C$ (except perhaps for $\omega$ in some null set), or
2. there exists $C$ such that if $n\leq T$ holds for all $\omega$, then $|X_n|\leq C$ holds a.s.

I don't think the second alternative is enough to imply that $X_T$ is integrable, but I would appreciate a clarification here.