Prove that, if $(X_n)_{n\in\mathbb{N}}$ is martingale terms of filtration $(\mathcal{F}_n)_{n\in\mathbb{N}}$ that $\max\{X_n,a\},a\in\mathbb{R}$ is a submartingale.
I've done so far
$$\mathbb{E}(\max\{X_{n+1},a\}|\mathcal{F}_n)=\left\{\begin{array}{ll} \mathbb{E}(a|\mathcal{F}_n),&a\geq X_{n+1}\\ \mathbb{E}(X_{n+1}|\mathcal{F}_n),&a<X_{n+1} \end{array}\right.=\left\{\begin{array}{ll} a,&a\geq X_{n+1}\\ X_n,&a<X_{n+1} \end{array}\right.$$
And I don't know what to do next. How should I show that it is greater or equal than $\max\{X_n,a\}$? Please help
Hint: conditional expectation is monotone.
Clearly $\max\{X_{n+1},a\} \ge a$. So $E[\max\{X_{n+1},a\} \mid \mathcal{F}_n] \ge \dots ?$
Likewise, $\max\{X_{n+1},a\} \ge X_{n+1}$. So $E[\max\{X_{n+1},a\} \mid \mathcal{F}_n] \ge \dots?$