Q1
I was so confused about this maximal sequence, I thought it was a constant random variable sequence at first, but then I realised it should be an increasing sequence (not really sure). Then I have a question about this, how could you guarantee the existence of such a sequence? I could convince myself that some terms $M_p>M_q$ for $0<p,q<n$ for some $\omega\in\Omega$, how could you define one term of $(M_n)_{n\in\mathbb{N}}$ is greater than another for all $\omega\in\Omega$?
Q2
I don't quite get the final inequality, I know $E(M_n\mathbf{1}_{\{M_n\geq\alpha\}})\leq E(M_n)$ holds, and I proposed $E(M_n\mathbf{1}_{\{M^*_n\geq\alpha\}})\leq E(M_n\mathbf{1}_{\{M_n\geq\alpha\}})$, but I don't know how to prove this.
Thank you for your help
I think Q1 was answered in your other question.
For Q2, this is just because $(M_n)$ is non-negative. Therefore $M_n 1_{M_n^* \ge \alpha} \le M_n$ - they're obviously equal if $M_n^* \ge \alpha$, and if $M_n^* < \alpha$, the LHS is $0$, which is automatically less than or equal to $M_n$. Then we just take expected value of both sides to get $\mathbb{E}[M_n 1_{M_n^* \ge \alpha}] \le \mathbb{E}[M_n]$,