If $Y_n=\min\{M_n,7\}$ and $\{M_n\}$ is a martingale wrt ${X_n}$, show that ${Y_n}$ is a supermartingale wrt ${X_n}$

Question

If $Y_n=\min\{M_n,7\}$ and $\{M_n\}$ is a martingale wrt ${X_n}$, show that ${Y_n}$ is a supermartingale wrt ${X_n}$

I tried doing cases for $M_n<7$ and for $M_n>7$, but I couldn't get that $E[M_{n+1}-M_n|X_{\le n}]\le 0$

Answer 1

The function $f(x)=\min\{x,7\}$ is concave, hence by Jensen's inequality for conditional expectation $$ \mathbb{E}[Y_{n+1}|\mathcal{F}_n]=\mathbb{E}[f(X_{n+1})|\mathcal{F}_n]\leq f(\mathbb{E}[X_{n+1}|\mathcal{F}_n])=f(X_n)=Y_n$$