I have a question regarding right-continuous super martingales. Let $(X_t, \mathcal{F}_t)_{t\geq0}$ be a right continuous super-martingale, and $\tau$ a $(\mathcal{F}_t)_{t\geq0}$-stopping time. I want to show that both $(X_{t \land \tau}, \mathcal{F}_{t \land \tau})_{t \geq0}$ and $(X_{t \land \tau}, \mathcal{F}_{t})_{t \geq0}$ are super-martingales.
I have the above result for discrete super-martingales, with respect to non-shifted filtration. That is, I know that if $(M_n, \mathcal{F}_n)_{n \in \mathbb{N}}$ is a super-martingale, $\tau$ a $(\mathcal{F}_n)_{n \in \mathbb{N}}$-stopping time then $(M_{n \land \tau}, \mathcal{F}_{n})_{n \in \mathbb{N}}$ is a super-martingale. I am not sure how can I generalize this to shifted filtration and continuous time. I couldn't find a proper answer or reference to that. Any help will be greatly appreciated.