In my lecture notes they use the fact, that every supermartingale $(M_t)$ for which the map $t\mapsto E[M_t]$ is constant is already a martingale. Unfortunately I can't prove it. Some help would be appreciated.

By definition of a supermartingale we have: $E[M_0]\ge E[M_s]\ge E[M_t]$ for $t\ge s\ge 0$. I also know that $M_s\ge E[M_t|\mathcal{F}_s]$. If I would take expectation I would get an equality by assumption. However I do not see how this helps here to prove $M_s= E[M_t|\mathcal{F}_s]$