Doob's Maximal Inequality/Discrete Time/Proof 2

Proof
Let $\lambda > 0 $ and $n \ge 0$.

Let:
 * $E := \set {X^\ast _n \ge \lambda}$.

That is, $E$ is a disjoint union:
 * $(1):\quad \ds E = \bigsqcup _{0 \mathop \le k \mathop \le n} E_k$

where:
 * $\ds E_k := \set {X_k \ge \lambda} \cap \bigcap _{0 \mathop \le j \mathop \le k-1} \set {X_j < \lambda}$

By construction, we have:
 * $\forall k \in \set {0, \ldots, n} : E_k \in \FF_k$

In particular:

where $\chi_A$ denotes the characteristic function of $A \subseteq \Omega$.

Therefore: