I am having trouble proving the following statement:
Let $\alpha>0$ and $M:\Omega\times [0,1]\longrightarrow \mathbb{R}$ be a super martingale. Then $M\wedge \alpha$ is also a super martingale.
The adaptedness/measurability and the integrability requirements are easy. But I am struggling to show that
\begin{equation*} \mathbb{E}[M_t\wedge \alpha | \mathscr{F}_s] \le M_s\wedge \alpha \qquad \forall s<t \end{equation*}
Any help would be greatly appreciated! Thanks in advanced