Let $(X_n)_{n\geq0}$ be a non-negative supermatingale and $T = \inf\{n \geq 0 : X_n = 0\}$. Show that on the event $\{T < \infty\}$, $X_{T+n} = 0$ for all $n \geq 0$ a.s.
My approach: $0 \leq E[X_{T+n}|\mathcal{F}_{T+n-1}] \leq X_{T+n-1} \leq \cdots \leq X_T = 0$. So, $X_{T+n} = 0$ for all $n$. Is this correct?
$X_n$ is uniformly bounded from below, so it converges a.s. to an integrable random variable for Doob's Supermartingale Convergence Theorem.
This fact together with the event $\{T < \infty\}$ allows us to use the Optional Stopping Theorem, so
$\mathsf{E}[X_{T+n}|\mathscr{F}_{T}]\le X_{T}=0$.
But since $X$ is also bounded below, and is non negative, we have $\forall n\ge 1$
$0 \le \mathsf{E}[X_{T+n}|\mathscr{F}_{T}]\le X_{T}=0$.