Doob's continuous martingale convergence theorem

Question

I present the theorem and the part of the proof where I am having trouble understanding. This is a from lecture note I found online.

Here $\mathcal{D}_n=\{k/2^n|k\leq 0\}$, $\mathcal{D}=\cup_n \mathcal{D}_n$. I don't understand how Fatou's lemma was used to get the second inequality. Any help is appreciated. Thanks!

Answer 1

Let $A_n:=\left\{\sup_{t\in\mathcal D_n,t\leqslant n}\left\lvert M_t\right\rvert \gt \lambda\right\}$. After the application of Doob's inequality, we have $$ \mathbb P\left(A_n\right)\leqslant\frac 1\lambda \mathbb E\left\lvert M_n\right\rvert\leqslant \frac 1\lambda \sup_t\mathbb E\left\lvert M_t\right\rvert. $$ The sequence $\left(A_n\right)$ is non-increasing and $\bigcup_{n\geqslant 1}A_n$ is exactly the event $\left\{\sup_{t\in\mathcal D}\left\lvert M_t\right\rvert \gt \lambda\right\}$.