I am trying to prove that if I have a right-continuous supermartingale $(S_t,\mathcal{F}_t)_{t\geq0}$ and $\tau <\infty$ a stopping time, that $(S_{\tau \wedge t},\mathcal{F}_{\tau \wedge t})_{t\geq0}$ is also supermartingale.
I proved it for the discrete time case but I don't know how I can take the limit and prove it for the continuous time case.
I tried to take $\tau_n \searrow \tau$ and show uniform integrabillity of $S_{\tau_n \wedge t}$ but I am stuck.
Thank you