Let $X_t \in L^{1}(\mathbb{P})$ be an adapted process $\forall t$. Show that the following definitions are equivalent :
(1) $X$ is a $\mathbb{P}$-supermartingale, i.e. $X_s \geq \mathbb{E}_{\mathbb{P}}[X_t|F_s]$ for $0 \leq s \leq t \ \leq T$.
(2) $X_{t-1} \geq \mathbb{E}_{\mathbb{P}}[X_t | F_{t-1}]$ $1 \leq t \leq T$
My problem is I do not know where to start and how to show this in a precise way (don't they follow immediately from each other?) and how can I make use of the fact that $X_t \in L^{1}(\mathbb{P})$?